המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי

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</div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he" class=""><span>תרומה לוויקיפדיה</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%9E%D7%A9%D7%A4%D7%98+%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99+%D7%A9%D7%9C+%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F+%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99+%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%9E%D7%A9%D7%A4%D7%98+%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99+%D7%A9%D7%9C+%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F+%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99+%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%9E%D7%A9%D7%A4%D7%98+%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99+%D7%A9%D7%9C+%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F+%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99+%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%9E%D7%A9%D7%A4%D7%98+%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99+%D7%A9%D7%9C+%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F+%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99+%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv 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class="vector-pinnable-header-label">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">התחלה</div> </a> </li> <li id="toc-ניסוח_פורמלי" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ניסוח_פורמלי"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>ניסוח פורמלי</span> </div> </a> <button aria-controls="toc-ניסוח_פורמלי-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק ניסוח פורמלי</span> </button> <ul id="toc-ניסוח_פורמלי-sublist" class="vector-toc-list"> <li id="toc-משפט" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#משפט"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>משפט</span> </div> </a> <ul id="toc-משפט-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-נוסחת_ניוטון-לייבניץ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#נוסחת_ניוטון-לייבניץ"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>נוסחת ניוטון-לייבניץ</span> </div> </a> <ul id="toc-נוסחת_ניוטון-לייבניץ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-הוכחה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הוכחה"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>הוכחה</span> </div> </a> <button aria-controls="toc-הוכחה-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק הוכחה</span> </button> <ul id="toc-הוכחה-sublist" class="vector-toc-list"> <li id="toc-הפונקציה_F_רציפה" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הפונקציה_F_רציפה"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>הפונקציה F רציפה</span> </div> </a> <ul id="toc-הפונקציה_F_רציפה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הפונקציה_f_היא_נגזרת_של_F_בנקודות_הרציפות_שלה" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הפונקציה_f_היא_נגזרת_של_F_בנקודות_הרציפות_שלה"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>הפונקציה f היא נגזרת של F בנקודות הרציפות שלה</span> </div> </a> <ul id="toc-הפונקציה_f_היא_נגזרת_של_F_בנקודות_הרציפות_שלה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קיום_פונקציה_קדומה_בקטע_ונוסחת_ניוטון-לייבניץ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#קיום_פונקציה_קדומה_בקטע_ונוסחת_ניוטון-לייבניץ"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>קיום פונקציה קדומה בקטע ונוסחת ניוטון-לייבניץ</span> </div> </a> <ul id="toc-קיום_פונקציה_קדומה_בקטע_ונוסחת_ניוטון-לייבניץ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הוכחה_לנוסחת_ניוטון-לייבניץ_שאינה_מתבססת_על_המשפט_היסודי" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הוכחה_לנוסחת_ניוטון-לייבניץ_שאינה_מתבססת_על_המשפט_היסודי"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>הוכחה לנוסחת ניוטון-לייבניץ שאינה מתבססת על המשפט היסודי</span> </div> </a> <ul id="toc-הוכחה_לנוסחת_ניוטון-לייבניץ_שאינה_מתבססת_על_המשפט_היסודי-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-הכללות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הכללות"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>הכללות</span> </div> </a> <ul id="toc-הכללות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="תוכן עניינים" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="מצב תוכן העניינים" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">מצב תוכן העניינים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="מעבר לערך בשפה אחרת. זמין ב־56 שפות" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">56 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus – אנגלית" lang="en" hreflang="en" data-title="Fundamental theorem of calculus" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%AB%E1%88%8D%E1%8A%A9%E1%88%88%E1%88%B5_%E1%88%98%E1%88%B0%E1%88%A8%E1%89%B3%E1%8B%8A_%E1%8A%A5%E1%88%AD%E1%8C%8D%E1%8C%A5" title="የካልኩለስ መሰረታዊ እርግጥ – אמהרית" lang="am" hreflang="am" data-title="የካልኩለስ መሰረታዊ እርግጥ" data-language-autonym="አማርኛ" data-language-local-name="אמהרית" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A7%D9%84%D8%A3%D8%B3%D8%A7%D8%B3%D9%8A%D8%A9_%D9%84%D9%84%D8%AA%D9%81%D8%A7%D8%B6%D9%84_%D9%88%D8%A7%D9%84%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="المبرهنة الأساسية للتفاضل والتكامل – ערבית" lang="ar" hreflang="ar" data-title="المبرهنة الأساسية للتفاضل والتكامل" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teorema_fundamental_del_c%C3%A1lculu" title="Teorema fundamental del cálculu – אסטורית" lang="ast" hreflang="ast" data-title="Teorema fundamental del cálculu" data-language-autonym="Asturianu" data-language-local-name="אסטורית" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD-%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Ньютон-Лейбниц теоремаһы – בשקירית" lang="ba" hreflang="ba" data-title="Ньютон-Лейбниц теоремаһы" data-language-autonym="Башҡортса" data-language-local-name="בשקירית" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9D%D1%8C%D1%8E%D1%82%D0%B0%D0%BD%D0%B0_%E2%80%94_%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D1%96%D1%86%D0%B0" title="Формула Ньютана — Лейбніца – בלארוסית" lang="be" hreflang="be" data-title="Формула Ньютана — Лейбніца" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%B4%D0%B0%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D0%BB%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Фундаментална теорема на анализа – בולגרית" lang="bg" hreflang="bg" data-title="Фундаментална теорема на анализа" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B2%E0%A6%95%E0%A7%81%E0%A6%B2%E0%A6%BE%E0%A6%B8%E0%A7%87%E0%A6%B0_%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="ক্যালকুলাসের মৌলিক উপপাদ্য – בנגלית" lang="bn" hreflang="bn" data-title="ক্যালকুলাসের মৌলিক উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_fonamental_del_c%C3%A0lcul" title="Teorema fonamental del càlcul – קטלאנית" lang="ca" hreflang="ca" data-title="Teorema fonamental del càlcul" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%D8%A8%D9%86%DB%95%DA%95%DB%95%D8%AA%DB%8C%DB%8C_%DA%A9%D8%A7%D9%84%DA%A9%DB%8C%D9%88%D9%84%D8%B3" title="تیۆرمی بنەڕەتیی کالکیولس – כורדית סוראנית" lang="ckb" hreflang="ckb" data-title="تیۆرمی بنەڕەتیی کالکیولس" data-language-autonym="کوردی" data-language-local-name="כורדית סוראנית" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Z%C3%A1kladn%C3%AD_v%C4%9Bta_integr%C3%A1ln%C3%ADho_po%C4%8Dtu" title="Základní věta integrálního počtu – צ׳כית" lang="cs" hreflang="cs" data-title="Základní věta integrálního počtu" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%E2%80%94%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B8" title="Ньютон—Лейбниц формули – צ׳ובשית" lang="cv" hreflang="cv" data-title="Ньютон—Лейбниц формули" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Infinitesimalregningens_hoveds%C3%A6tning" title="Infinitesimalregningens hovedsætning – דנית" lang="da" hreflang="da" data-title="Infinitesimalregningens hovedsætning" data-language-autonym="Dansk" data-language-local-name="דנית" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="ערך טוב"><a href="https://de.wikipedia.org/wiki/Fundamentalsatz_der_Analysis" title="Fundamentalsatz der Analysis – גרמנית" lang="de" hreflang="de" data-title="Fundamentalsatz der Analysis" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CE%BC%CE%B5%CE%BB%CE%B9%CF%8E%CE%B4%CE%B5%CF%82_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CE%BF%CF%8D" title="Θεμελιώδες θεώρημα του λογισμού – יוונית" lang="el" hreflang="el" data-title="Θεμελιώδες θεώρημα του λογισμού" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Fundamenta_teoremo_de_kalkulo" title="Fundamenta teoremo de kalkulo – אספרנטו" lang="eo" hreflang="eo" data-title="Fundamenta teoremo de kalkulo" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_fundamental_del_c%C3%A1lculo" title="Teorema fundamental del cálculo – ספרדית" lang="es" hreflang="es" data-title="Teorema fundamental del cálculo" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kalkuluaren_oinarrizko_teorema" title="Kalkuluaren oinarrizko teorema – בסקית" lang="eu" hreflang="eu" data-title="Kalkuluaren oinarrizko teorema" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A7%D8%B3%D8%A7%D8%B3%DB%8C_%D8%AD%D8%B3%D8%A7%D8%A8%D8%A7%D9%86" title="قضیه اساسی حسابان – פרסית" lang="fa" hreflang="fa" data-title="قضیه اساسی حسابان" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Analyysin_peruslause" title="Analyysin peruslause – פינית" lang="fi" hreflang="fi" data-title="Analyysin peruslause" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_fondamental_de_l%27analyse" title="Théorème fondamental de l'analyse – צרפתית" lang="fr" hreflang="fr" data-title="Théorème fondamental de l'analyse" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Bunteoirim_an_chalcalais" title="Bunteoirim an chalcalais – אירית" lang="ga" hreflang="ga" data-title="Bunteoirim an chalcalais" data-language-autonym="Gaeilge" data-language-local-name="אירית" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_fundamental_do_c%C3%A1lculo" title="Teorema fundamental do cálculo – גליסית" lang="gl" hreflang="gl" data-title="Teorema fundamental do cálculo" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8_%E0%A4%95%E0%A4%BE_%E0%A4%AE%E0%A5%82%E0%A4%B2%E0%A4%AD%E0%A5%82%E0%A4%A4_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="कलन का मूलभूत प्रमेय – הינדי" lang="hi" hreflang="hi" data-title="कलन का मूलभूत प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Newton%E2%80%93Leibniz-t%C3%A9tel" title="Newton–Leibniz-tétel – הונגרית" lang="hu" hreflang="hu" data-title="Newton–Leibniz-tétel" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus – אינדונזית" lang="id" hreflang="id" data-title="Teorema dasar kalkulus" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_fondamentale_del_calcolo_integrale" title="Teorema fondamentale del calcolo integrale – איטלקית" lang="it" hreflang="it" data-title="Teorema fondamentale del calcolo integrale" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E7%A9%8D%E5%88%86%E5%AD%A6%E3%81%AE%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="微分積分学の基本定理 – יפנית" lang="ja" hreflang="ja" data-title="微分積分学の基本定理" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%90%E1%83%9A%E1%83%99%E1%83%A3%E1%83%9A%E1%83%A3%E1%83%A1%E1%83%98%E1%83%A1_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%93%E1%83%90%E1%83%9B%E1%83%94%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%94%E1%83%91%E1%83%98" title="კალკულუსის ფუნდამენტური თეორემები – גאורגית" lang="ka" hreflang="ka" data-title="კალკულუსის ფუნდამენტური თეორემები" data-language-autonym="ქართული" data-language-local-name="גאורגית" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86-%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Лейбниц-Ньютон теоремасы – קזחית" lang="kk" hreflang="kk" data-title="Лейбниц-Ньютон теоремасы" data-language-autonym="Қазақша" data-language-local-name="קזחית" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B2%A6_%E0%B2%AE%E0%B3%82%E0%B2%B2%E0%B2%AD%E0%B3%82%E0%B2%A4_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B3%87%E0%B2%AF" title="ಕಲನಶಾಸ್ತ್ರದ ಮೂಲಭೂತ ಪ್ರಮೇಯ – קנאדה" lang="kn" hreflang="kn" data-title="ಕಲನಶಾಸ್ತ್ರದ ಮೂಲಭೂತ ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="קנאדה" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EC%A0%95%EB%A6%AC" title="미적분학의 기본 정리 – קוריאנית" lang="ko" hreflang="ko" data-title="미적분학의 기본 정리" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theorema_fundamentale_calculi" title="Theorema fundamentale calculi – לטינית" lang="la" hreflang="la" data-title="Theorema fundamentale calculi" data-language-autonym="Latina" data-language-local-name="לטינית" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_fondamental_del_calcol_integral" title="Teorema fondamental del calcol integral – לומברדית" lang="lmo" hreflang="lmo" data-title="Teorema fondamental del calcol integral" data-language-autonym="Lombard" data-language-local-name="לומברדית" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0%D1%82%D0%B0" title="Основна теорема на анализата – מקדונית" lang="mk" hreflang="mk" data-title="Основна теорема на анализата" data-language-autonym="Македонски" data-language-local-name="מקדונית" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teorem_asas_kalkulus" title="Teorem asas kalkulus – מלאית" lang="ms" hreflang="ms" data-title="Teorem asas kalkulus" data-language-autonym="Bahasa Melayu" data-language-local-name="מלאית" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoofdstelling_van_de_integraalrekening" title="Hoofdstelling van de integraalrekening – הולנדית" lang="nl" hreflang="nl" data-title="Hoofdstelling van de integraalrekening" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Analysens_fundamentalteorem" title="Analysens fundamentalteorem – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Analysens fundamentalteorem" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podstawowe_twierdzenie_rachunku_ca%C5%82kowego" title="Podstawowe twierdzenie rachunku całkowego – פולנית" lang="pl" hreflang="pl" data-title="Podstawowe twierdzenie rachunku całkowego" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_fundamental_do_c%C3%A1lculo" title="Teorema fundamental do cálculo – פורטוגזית" lang="pt" hreflang="pt" data-title="Teorema fundamental do cálculo" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_fundamental%C4%83_a_calculului_integral" title="Teorema fundamentală a calculului integral – רומנית" lang="ro" hreflang="ro" data-title="Teorema fundamentală a calculului integral" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0_%E2%80%94_%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86%D0%B0" title="Теорема Ньютона — Лейбница – רוסית" lang="ru" hreflang="ru" data-title="Теорема Ньютона — Лейбница" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Fundamental theorem of calculus" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Z%C3%A1kladn%C3%A1_veta_diferenci%C3%A1lneho_a_integr%C3%A1lneho_po%C4%8Dtu" title="Základná veta diferenciálneho a integrálneho počtu – סלובקית" lang="sk" hreflang="sk" data-title="Základná veta diferenciálneho a integrálneho počtu" data-language-autonym="Slovenčina" data-language-local-name="סלובקית" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Osnovni_izrek_infinitezimalnega_ra%C4%8Duna" title="Osnovni izrek infinitezimalnega računa – סלובנית" lang="sl" hreflang="sl" data-title="Osnovni izrek infinitezimalnega računa" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Fundamentalna_teorema_ra%C4%8Duna" title="Fundamentalna teorema računa – סרבית" lang="sr" hreflang="sr" data-title="Fundamentalna teorema računa" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Analysens_fundamentalsats" title="Analysens fundamentalsats – שוודית" lang="sv" hreflang="sv" data-title="Analysens fundamentalsats" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%A1%E0%B8%B9%E0%B8%A5%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%81%E0%B8%84%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%A5%E0%B8%B1%E0%B8%AA" title="ทฤษฎีบทมูลฐานของแคลคูลัส – תאית" lang="th" hreflang="th" data-title="ทฤษฎีบทมูลฐานของแคลคูลัส" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kalk%C3%BCl%C3%BCs%C3%BCn_temel_teoremi" title="Kalkülüsün temel teoremi – טורקית" lang="tr" hreflang="tr" data-title="Kalkülüsün temel teoremi" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD-%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D1%81%D1%8B" title="Ньютон-Лейбниц формуласы – טטרית" lang="tt" hreflang="tt" data-title="Ньютон-Лейбниц формуласы" data-language-autonym="Татарча / tatarça" data-language-local-name="טטרית" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0_%E2%80%94_%D0%9B%D1%8F%D0%B9%D0%B1%D0%BD%D1%96%D1%86%D0%B0" title="Формула Ньютона — Ляйбніца – אוקראינית" lang="uk" hreflang="uk" data-title="Формула Ньютона — Ляйбніца" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8%D8%A7%D9%86_%DA%A9%D8%A7_%D8%A8%D9%86%DB%8C%D8%A7%D8%AF%DB%8C_%D9%82%D8%B6%DB%8C%DB%81" title="حسابان کا بنیادی قضیہ – אורדו" lang="ur" hreflang="ur" data-title="حسابان کا بنیادی قضیہ" data-language-autonym="اردو" data-language-local-name="אורדו" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_c%C6%A1_b%E1%BA%A3n_c%E1%BB%A7a_gi%E1%BA%A3i_t%C3%ADch" title="Định lý cơ bản của giải tích – וייטנאמית" lang="vi" hreflang="vi" data-title="Định lý cơ bản của giải tích" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="微积分基本定理 – סינית" lang="zh" hreflang="zh" data-title="微积分基本定理" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%AE-chek-hun_%C3%AA_ki-p%C3%BAn_t%C4%93ng-l%C3%AD" title="Bî-chek-hun ê ki-pún tēng-lí – מין נאנית" lang="nan" hreflang="nan" data-title="Bî-chek-hun ê ki-pún tēng-lí" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="מין נאנית" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="微積分基本定理 – קנטונזית" lang="yue" hreflang="yue" data-title="微積分基本定理" data-language-autonym="粵語" data-language-local-name="קנטונזית" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1217677#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="מרחבי שם"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="צפייה בדף התוכן [c]" accesskey="c"><span>ערך</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D7%A9%D7%99%D7%97%D7%94:%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" rel="discussion" title="שיחה על דף התוכן [t]" accesskey="t"><span>שיחה</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="שינוי הגוון השפה" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">עברית</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="צפיות"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99"><span>קריאה</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit" title="עריכת קוד המקור של הדף הזה [e]" accesskey="e"><span>עריכת קוד מקור</span></a></li><li id="ca-ve-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit" title="עריכת הדף הזה [v]" accesskey="v"><span>עריכה</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=history" title="גרסאות קודמות של דף זה [h]" accesskey="h"><span>גרסאות קודמות</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="כלי דף"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="כלים" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">כלים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">כלים</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">הסתרה</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="אפשרויות נוספות" > <div class="vector-menu-heading"> פעולות </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99"><span>קריאה</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit" title="עריכת קוד המקור של הדף הזה [e]" accesskey="e"><span>עריכת קוד מקור</span></a></li><li id="ca-more-ve-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit" title="עריכת הדף הזה [v]" accesskey="v"><span>עריכה</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=history"><span>גרסאות קודמות</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> כללי </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D_%D7%9C%D7%9B%D7%90%D7%9F/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="רשימה של כל דפי הוויקי שמקשרים לדף הזה [j]" accesskey="j"><span>דפים המקושרים לכאן</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A9%D7%99%D7%A0%D7%95%D7%99%D7%99%D7%9D_%D7%91%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" rel="nofollow" title="השינויים האחרונים בדפים המקושרים מהדף הזה [k]" accesskey="k"><span>שינויים בדפים המקושרים</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%9E%D7%99%D7%95%D7%97%D7%93%D7%99%D7%9D" title="רשימה של כל הדפים המיוחדים [q]" accesskey="q"><span>דפים מיוחדים</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&oldid=39069661" title="קישור קבוע לגרסה הזאת של הדף הזה"><span>קישור קבוע</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=info" title="מידע נוסף על הדף הזה"><span>מידע על הדף</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A6%D7%99%D7%98%D7%95%D7%98_%D7%93%D7%A3_%D7%96%D7%94&page=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&id=39069661&wpFormIdentifier=titleform" title="מידע איך לצטט את הדף הזה"><span>ציטוט הדף הזה</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9E%D7%A7%D7%A6%D7%A8_%D7%9B%D7%AA%D7%95%D7%91%D7%95%D7%AA&url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2594%25D7%259E%25D7%25A9%25D7%25A4%25D7%2598_%25D7%2594%25D7%2599%25D7%25A1%25D7%2595%25D7%2593%25D7%2599_%25D7%25A9%25D7%259C_%25D7%2594%25D7%2597%25D7%25A9%25D7%2591%25D7%2595%25D7%259F_%25D7%2594%25D7%2593%25D7%2599%25D7%25A4%25D7%25A8%25D7%25A0%25D7%25A6%25D7%2599%25D7%2590%25D7%259C%25D7%2599_%25D7%2595%25D7%2594%25D7%2590%25D7%2599%25D7%25A0%25D7%2598%25D7%2592%25D7%25A8%25D7%259C%25D7%2599"><span>קבלת כתובת מקוצרת</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:QrCode&url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2594%25D7%259E%25D7%25A9%25D7%25A4%25D7%2598_%25D7%2594%25D7%2599%25D7%25A1%25D7%2595%25D7%2593%25D7%2599_%25D7%25A9%25D7%259C_%25D7%2594%25D7%2597%25D7%25A9%25D7%2591%25D7%2595%25D7%259F_%25D7%2594%25D7%2593%25D7%2599%25D7%25A4%25D7%25A8%25D7%25A0%25D7%25A6%25D7%2599%25D7%2590%25D7%259C%25D7%2599_%25D7%2595%25D7%2594%25D7%2590%25D7%2599%25D7%25A0%25D7%2598%25D7%2592%25D7%25A8%25D7%259C%25D7%2599"><span>הורדת קוד QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> הדפסה/יצוא </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A1%D7%A4%D7%A8&bookcmd=book_creator&referer=%D7%94%D7%9E%D7%A9%D7%A4%D7%98+%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99+%D7%A9%D7%9C+%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F+%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99+%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99"><span>יצירת ספר</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:DownloadAsPdf&page=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=show-download-screen"><span>הורדה כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a 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לפריט המשויך במאגר הנתונים [g]" accesskey="g"><span>פריט ויקינתונים</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="כלי דף"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="מראה"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><p><b>המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</b> או <b>המשפט היסודי של החשבון האינפיניטסימלי</b> הוא משפט מתמטי הקושר בין שני מושגי היסוד של <a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">החשבון האינפיניטסימלי</a>: ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> וה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל">אינטגרל</a>. המשפט מראה שגזירה ואינטגרציה הן פעולות הופכיות זו לזו: אם <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציה</a> <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" class="mw-redirect" title="רציפות">רציפה</a> עוברת אינטגרציה ואחר כך גוזרים את התוצאה, חוזרים לפונקציה המקורית. פרט לקשר זה, המשפט גם מספק שיטה מעשית לחישוב <a href="/wiki/%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%94%D7%9E%D7%A1%D7%95%D7%99%D7%9D" class="mw-redirect" title="האינטגרל המסוים">האינטגרל המסוים</a> (שהוא מושג המוגדר בצורה שאינה מאפשרת חישוב פשוט) באמצעות <a href="/wiki/%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%94%D7%9C%D7%90_%D7%9E%D7%A1%D7%95%D7%99%D7%9D" class="mw-redirect" title="האינטגרל הלא מסוים">האינטגרל הלא מסוים</a>, שלחישובו יש דרכים רבות יותר. </p><p>המשפט היסודי של החשבון האינפיניטסימלי קובע, שעבור פונקציות אינטגרביליות שיש להן פונקציה קדומה, האינטגרל המסוים בקטע כלשהו, שווה להפרש הערכים של האינטגרל הלא המסוים שלה בנקודות שבקצוות הקטע. </p><p>לכאורה שני מושגים אלה שונים ובאים מעולמות שונים, אבל המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי קובע את הקשר העמוק בין שני התחומים. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="ניסוח_פורמלי"><span id=".D7.A0.D7.99.D7.A1.D7.95.D7.97_.D7.A4.D7.95.D7.A8.D7.9E.D7.9C.D7.99"></span>ניסוח פורמלי</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=1" title="עריכת קוד המקור של הפרק: ניסוח פורמלי"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=1" title="עריכת פסקה: "ניסוח פורמלי"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:FTC_geometric.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/FTC_geometric.svg/400px-FTC_geometric.svg.png" decoding="async" width="400" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/FTC_geometric.svg/600px-FTC_geometric.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/FTC_geometric.svg/800px-FTC_geometric.svg.png 2x" data-file-width="627" data-file-height="362" /></a><figcaption>כאשר (A(x הוא ה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9E%D7%A1%D7%95%D7%99%D7%9D" class="mw-redirect" title="אינטגרל מסוים">אינטגרל המסוים</a> של (f(x, המגדיר את ה<a href="/wiki/%D7%A9%D7%98%D7%97" title="שטח">שטח</a> מתחת ל-f בין <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%94_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="נקודה (גאומטריה)">נקודה</a> קבועה a (במקרה הזה, a=0) לבין x כלשהו, המשפט היסודי קובע כי ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> של A שווה ל-f. בציור קל לראות ששטח המלבן האדום שווה מצד אחד לשינוי בשטח מתחת ל<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציה</a> (השינוי ב-A) ומצד שני שווה ב<a href="/wiki/%D7%A7%D7%99%D7%A8%D7%95%D7%91" title="קירוב">קירוב</a> (הולך ומשתפר עבור ערכי h קטנים) ל-f(x)*h. כאשר מחלקים ב-h ומשאיפים אותו ל<a href="/wiki/0_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="0 (מספר)">אפס</a>, מקבלים את הגדרת הנגזרת.</figcaption></figure> <p>המשפט היסודי של החשבון האינטגרלי מורכב בעצם משני משפטים: </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Fundamental_theorem_of_calculus_(animation).gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/2/2f/Fundamental_theorem_of_calculus_%28animation%29.gif" decoding="async" width="503" height="389" class="mw-file-element" data-file-width="503" data-file-height="389" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="משפט"><span id=".D7.9E.D7.A9.D7.A4.D7.98"></span>משפט</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=2" title="עריכת קוד המקור של הפרק: משפט"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=2" title="עריכת פסקה: "משפט"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%91%D7%99%D7%9C%D7%99%D7%AA" class="mw-redirect" title="פונקציה אינטגרבילית">פונקציה אינטגרבילית</a> בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> ויהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f41f5bd8787016205747827b2bf6624bd4f44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.029ex; height:5.843ex;" alt="{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span> <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9E%D7%A1%D7%95%D7%99%D7%9D" class="mw-redirect" title="אינטגרל מסוים">אינטגרל מסוים</a> שלה, כך ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.725ex; height:2.843ex;" alt="{\displaystyle x\in [a,b]}"></span>. אזי: </p> <ol><li>הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94" class="mw-redirect" title="פונקציה רציפה">רציפה</a>.</li> <li>בכל נקודה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> בה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> רציפה, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cda729f4fc92e3228240eb588757ff43659c974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.321ex; height:2.176ex;" alt="{\displaystyle \ F}"></span> <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">גזירה</a> ומתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(x_{0})=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x_{0})=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bc24e9d2287452424fe2c8181a07da65bdc8cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.263ex; height:3.009ex;" alt="{\displaystyle F'(x_{0})=f(x_{0})}"></span> .</li> <li>אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> רציפה בכל הקטע, אזי קיימת לה <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%93%D7%95%D7%9E%D7%94" class="mw-redirect" title="פונקציה קדומה">פונקציה קדומה</a> בקטע, והפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24411c18e9fb6a78762e27868d58cf4fbf0a883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.449ex; height:5.843ex;" alt="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span> היא <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%93%D7%95%D7%9E%D7%94" class="mw-redirect" title="פונקציה קדומה">פונקציה קדומה</a> שמקיימת <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10131b70a7b0b4c39667f5386b14c449a5217e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle F'=f}"></span> בכל הקטע.</li></ol> <p>יתרה מזאת, לכל <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A2_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="קבוע מתמטי">קבוע</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5e9d92f85e372ffc7e6736ac26ea8fef1d473c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.055ex; height:5.843ex;" alt="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t+C}"></span> פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="נוסחת_ניוטון-לייבניץ"><span id=".D7.A0.D7.95.D7.A1.D7.97.D7.AA_.D7.A0.D7.99.D7.95.D7.98.D7.95.D7.9F-.D7.9C.D7.99.D7.99.D7.91.D7.A0.D7.99.D7.A5"></span>נוסחת ניוטון-לייבניץ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=3" title="עריכת קוד המקור של הפרק: נוסחת ניוטון-לייבניץ"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=3" title="עריכת פסקה: "נוסחת ניוטון-לייבניץ"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> פונקציה אינטגרבילית שיש לה פונקציה קדומה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cda729f4fc92e3228240eb588757ff43659c974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.321ex; height:2.176ex;" alt="{\displaystyle \ F}"></span> בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c40bbeaa2f59e60b6259cebe2479bc24396f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.136ex; height:2.843ex;" alt="{\displaystyle \ [a,b]}"></span>. אם נסמן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{\mathrm {d} x}}F(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{\mathrm {d} x}}F(x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b42507bc9ca26a974759016b44b6d58d72d3383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.854ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{\mathrm {d} x}}F(x)=f(x)}"></span> אזי </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \int _{a}^{b}f(x)\,\mathrm {d} x=F(x)\mid _{a}^{b}=F(b)-F(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \int _{a}^{b}f(x)\,\mathrm {d} x=F(x)\mid _{a}^{b}=F(b)-F(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f262cfeaf60b5adcb7dd8a057e2d577b496157a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.177ex; height:6.343ex;" alt="{\displaystyle \ \int _{a}^{b}f(x)\,\mathrm {d} x=F(x)\mid _{a}^{b}=F(b)-F(a)}"></span></dd></dl> <p>נשים לב שאין חשיבות לשאלה איזו פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> לוקחים, מכיוון שכל הפונקציות הקדומות של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> נבדלות זו מזו בקבוע, והוא מתבטל כאשר מחשבים את ההפרש בין ערכי הפונקציה הקדומה בשתי נקודות שונות. </p><p>הנוסחה היסודית של החשבון האינפיניטסימלי מאפשרת לחשב אינטגרלים מסוימים של פונקציות מסוג מסוים. </p><p>ב<a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> מוכללת נוסחה זו למשפחה רחבה יותר של פונקציות, הפונקציות <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA_%D7%91%D7%94%D7%97%D7%9C%D7%98" class="mw-redirect" title="רציפות בהחלט">הרציפות בהחלט</a>. ניתן להראות גם שזו משפחת הפונקציות הרחבה ביותר עבורה מתקיימת נוסחה זו. ישנן פונקציות רציפות וגזירות <a href="/wiki/%D7%9B%D7%9E%D7%A2%D7%98_%D7%91%D7%9B%D7%9C_%D7%9E%D7%A7%D7%95%D7%9D" class="mw-redirect" title="כמעט בכל מקום">כמעט בכל מקום</a> (אבל לא בכל מקום) שאינן האינטגרל של נגזרתן (ראו <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A1%D7%99%D7%A0%D7%92%D7%95%D7%9C%D7%A8%D7%99%D7%AA" title="פונקציה סינגולרית">פונקציה סינגולרית</a>). </p> <div class="mw-heading mw-heading2"><h2 id="הוכחה"><span id=".D7.94.D7.95.D7.9B.D7.97.D7.94"></span>הוכחה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=4" title="עריכת קוד המקור של הפרק: הוכחה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=4" title="עריכת פסקה: "הוכחה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="הפונקציה_F_רציפה"><span id=".D7.94.D7.A4.D7.95.D7.A0.D7.A7.D7.A6.D7.99.D7.94_F_.D7.A8.D7.A6.D7.99.D7.A4.D7.94"></span>הפונקציה F רציפה</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=5" title="עריכת קוד המקור של הפרק: הפונקציה F רציפה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=5" title="עריכת פסקה: "הפונקציה F רציפה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נתון ש־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span> <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%99%D7%9E%D7%9F#אינטגרביליות_לפי_רימן" title="אינטגרל רימן">אינטגרבילית לפי רימן</a> ולכן היא <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%97%D7%A1%D7%95%D7%9E%D7%94" title="פונקציה חסומה">חסומה</a>, כלומר קיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a031fb4bd04e0e0b821e6c09f1d81ec98c51b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.961ex; height:2.176ex;" alt="{\textstyle M\in \mathbb {R} }"></span> כך ש־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|f(x)\right|\leq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|f(x)\right|\leq M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169bfa15510016e19ed995a9f243e16a75311aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.252ex; height:2.843ex;" alt="{\textstyle \left|f(x)\right|\leq M}"></span> לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\in \left[a,b\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\in \left[a,b\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57dd0ff54c7b502598662117f42d70d022d9a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.725ex; height:2.843ex;" alt="{\textstyle x\in \left[a,b\right]}"></span>. יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x,y\in \left[a,b\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x,y\in \left[a,b\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b777d6026af93eee7a98b62dde4fd4b0251fa9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.915ex; height:2.843ex;" alt="{\textstyle x,y\in \left[a,b\right]}"></span>, אזי מתקיים: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left|F(x)-F(y)\right|=\left|\int _{a}^{x}f(t)\,\mathrm {d} t-\int _{a}^{y}f(t)\,\mathrm {d} t\right|=\left|\int _{y}^{x}f(t)\,\mathrm {d} t\right|\leq \int _{y}^{x}\left|f(t)\right|\,\mathrm {d} t\leq M\left|x-y\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>≤</mo> <mi>M</mi> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left|F(x)-F(y)\right|=\left|\int _{a}^{x}f(t)\,\mathrm {d} t-\int _{a}^{y}f(t)\,\mathrm {d} t\right|=\left|\int _{y}^{x}f(t)\,\mathrm {d} t\right|\leq \int _{y}^{x}\left|f(t)\right|\,\mathrm {d} t\leq M\left|x-y\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfbb6fcd7aeab9797bd2a2dec3ddf8e9859cc868" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:85.397ex; height:6.509ex;" alt="{\displaystyle \ \left|F(x)-F(y)\right|=\left|\int _{a}^{x}f(t)\,\mathrm {d} t-\int _{a}^{y}f(t)\,\mathrm {d} t\right|=\left|\int _{y}^{x}f(t)\,\mathrm {d} t\right|\leq \int _{y}^{x}\left|f(t)\right|\,\mathrm {d} t\leq M\left|x-y\right|}"></span></dd></dl> <p>כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256dae10b9e3abb3592ff608e81c8bc324edce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\textstyle F}"></span> מקיימת את <a href="/wiki/%D7%AA%D7%A0%D7%90%D7%99_%D7%9C%D7%99%D7%A4%D7%A9%D7%99%D7%A5" title="תנאי ליפשיץ">תנאי ליפשיץ</a> ב־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left[a,b\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left[a,b\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51aebfe7a9a5b11ce857ef1c4457f3b3d209db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle \left[a,b\right]}"></span>, ולכן היא <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" class="mw-redirect" title="רציפות במידה שווה">רציפה במידה שווה</a> (וּבִפרט רציפה). </p> <div class="mw-heading mw-heading3"><h3 id="הפונקציה_f_היא_נגזרת_של_F_בנקודות_הרציפות_שלה"><span id=".D7.94.D7.A4.D7.95.D7.A0.D7.A7.D7.A6.D7.99.D7.94_f_.D7.94.D7.99.D7.90_.D7.A0.D7.92.D7.96.D7.A8.D7.AA_.D7.A9.D7.9C_F_.D7.91.D7.A0.D7.A7.D7.95.D7.93.D7.95.D7.AA_.D7.94.D7.A8.D7.A6.D7.99.D7.A4.D7.95.D7.AA_.D7.A9.D7.9C.D7.94"></span>הפונקציה f היא נגזרת של F בנקודות הרציפות שלה</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=6" title="עריכת קוד המקור של הפרק: הפונקציה f היא נגזרת של F בנקודות הרציפות שלה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=6" title="עריכת פסקה: "הפונקציה f היא נגזרת של F בנקודות הרציפות שלה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תהא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc90e7d80812c3da5fca9a998c7136337be7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.965ex; height:2.009ex;" alt="{\displaystyle \ x_{0}}"></span> נקודת <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" class="mw-redirect" title="רציפות">רציפות</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span>. אנו רוצים להראות כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F'(x_{0})=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F'(x_{0})=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366cedb8cca4101577aed2bdd53c9437119a5332" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.844ex; height:3.009ex;" alt="{\displaystyle \ F'(x_{0})=f(x_{0})}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f443865903073cff9c53cc8b3fa6302ea1c409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.89ex; height:5.843ex;" alt="{\displaystyle \ F=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span>. </p><p>בסימון פורמלי יותר: אנו רוצים להראות כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6795463ba3f2b5669de8efb3e88b627cafafcf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.061ex; height:5.843ex;" alt="{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}"></span>. </p><p>על פי ההגדרה ואדיטיביות האינטגרל המסוים, אנו יודעים שמתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {F(x_{0}+h)-F(x_{0})}{h}}={\frac {1}{h}}\left(\int _{a}^{x_{0}+h}f(t)\,\mathrm {d} t-\int _{a}^{x_{0}}f(t)\,\mathrm {d} t\right)={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {F(x_{0}+h)-F(x_{0})}{h}}={\frac {1}{h}}\left(\int _{a}^{x_{0}+h}f(t)\,\mathrm {d} t-\int _{a}^{x_{0}}f(t)\,\mathrm {d} t\right)={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4311d76d477514e12b971c82187d8e6881c680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:76.244ex; height:6.676ex;" alt="{\displaystyle \ {\frac {F(x_{0}+h)-F(x_{0})}{h}}={\frac {1}{h}}\left(\int _{a}^{x_{0}+h}f(t)\,\mathrm {d} t-\int _{a}^{x_{0}}f(t)\,\mathrm {d} t\right)={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(t)\,\mathrm {d} t}"></span>. </p><p>כמו כן מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x_{0})={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(x_{0})\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x_{0})={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(x_{0})\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91791b08315ff28a0f8c6e5a7f07c6fc945b39da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.785ex; height:6.676ex;" alt="{\displaystyle \ f(x_{0})={\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}f(x_{0})\,\mathrm {d} t}"></span>, שכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db77e8f09fa7fa335d1828a9218a0e0df8136b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.052ex; height:2.843ex;" alt="{\displaystyle \ f(x_{0})}"></span> היא <b>קבוע</b>, ולכן האינטגרל שלה על הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ [x_{0},x_{0}+h]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ [x_{0},x_{0}+h]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f062ac3337535bccee575ce1deed5acef49fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.855ex; height:2.843ex;" alt="{\displaystyle \ [x_{0},x_{0}+h]}"></span> הוא פשוט אורך הקטע כפול <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db77e8f09fa7fa335d1828a9218a0e0df8136b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.052ex; height:2.843ex;" alt="{\displaystyle \ f(x_{0})}"></span>. </p><p>לכן מתקיים, על פי <a href="/wiki/%D7%90%D7%99_%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%94%D7%9E%D7%A9%D7%95%D7%9C%D7%A9_%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" class="mw-redirect" title="אי שוויון המשולש האינטגרלי">אי שוויון המשולש האינטגרלי</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|=\left|{\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}\left(f(t)-f(x_{0})\right)\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|=\left|{\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}\left(f(t)-f(x_{0})\right)\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3f94aced014cc2a362acd218625449673996b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:93.056ex; height:6.843ex;" alt="{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|=\left|{\frac {1}{h}}\int _{x_{0}}^{x_{0}+h}\left(f(t)-f(x_{0})\right)\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|}"></span>. </p><p>נזכור כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span> <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%94" class="mw-redirect" title="רציפה">רציפה</a> בנקודה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc90e7d80812c3da5fca9a998c7136337be7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.965ex; height:2.009ex;" alt="{\displaystyle \ x_{0}}"></span>, ולכן עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916f113f304b0379353c1350a27b37ce11582b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle \ \varepsilon >0}"></span> כלשהו קיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c79a1646e2b28ee2d649a0b0072cc4491539b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.89ex; height:2.343ex;" alt="{\displaystyle \ \delta >0}"></span> כך ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |t-x_{0}|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |t-x_{0}|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93f9c8ecebf2ee19abc81316bfa3b7e1cfd29fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.085ex; height:2.843ex;" alt="{\displaystyle \ |t-x_{0}|<\delta }"></span> גורר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |f(t)-f(x_{0})|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |f(t)-f(x_{0})|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb486e72c9ce9cb1f5027232df384f6ccfb42a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.296ex; height:2.843ex;" alt="{\displaystyle \ |f(t)-f(x_{0})|<\varepsilon }"></span>. </p><p>אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 0<|h|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 0<|h|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd8e6d8172286fd9d7d624b845fd9e8eb7d66ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.621ex; height:2.843ex;" alt="{\displaystyle \ 0<|h|<\delta }"></span> אז לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ t\in [x_{0},x_{0}+h]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ t\in [x_{0},x_{0}+h]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781f1a64243d0bc8ad81b275cf1f503fe4010d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.536ex; height:2.843ex;" alt="{\displaystyle \ t\in [x_{0},x_{0}+h]}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |t-x_{0}|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |t-x_{0}|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93f9c8ecebf2ee19abc81316bfa3b7e1cfd29fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.085ex; height:2.843ex;" alt="{\displaystyle \ |t-x_{0}|<\delta }"></span>. לכן: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\varepsilon \,\mathrm {d} t\right|={\frac {1}{|h|}}\varepsilon \cdot |h|=\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mi>ε</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>ε</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\varepsilon \,\mathrm {d} t\right|={\frac {1}{|h|}}\varepsilon \cdot |h|=\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3c4ee4e82259fb148389756f5fd03d48a4e9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.326ex; height:6.843ex;" alt="{\displaystyle \ {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\left|f(t)-f(x_{0})\right|\,\mathrm {d} t\right|\leq {\frac {1}{|h|}}\left|\int _{x_{0}}^{x_{0}+h}\varepsilon \,\mathrm {d} t\right|={\frac {1}{|h|}}\varepsilon \cdot |h|=\varepsilon }"></span>. </p><p>כלומר, הראינו כי לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916f113f304b0379353c1350a27b37ce11582b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle \ \varepsilon >0}"></span> ניתן למצוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c79a1646e2b28ee2d649a0b0072cc4491539b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.89ex; height:2.343ex;" alt="{\displaystyle \ \delta >0}"></span> כך שלכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |h|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |h|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6558913ef7ba8644a44a5e3ead81fa836f8ffa30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.36ex; height:2.843ex;" alt="{\displaystyle \ |h|<\delta }"></span> יתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a784c682989d101e6cbb38d084f8c6f901eae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.479ex; height:6.509ex;" alt="{\displaystyle \ \left|{\frac {F(x_{0}+h)-F(x_{0})}{h}}-f(x_{0})\right|<\varepsilon }"></span>, כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6795463ba3f2b5669de8efb3e88b627cafafcf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.061ex; height:5.843ex;" alt="{\displaystyle \ \lim _{h\rightarrow 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}=f(x_{0})}"></span>, כמבוקש. </p><p><b>מש"ל</b> </p> <div class="mw-heading mw-heading3"><h3 id="קיום_פונקציה_קדומה_בקטע_ונוסחת_ניוטון-לייבניץ"><span id=".D7.A7.D7.99.D7.95.D7.9D_.D7.A4.D7.95.D7.A0.D7.A7.D7.A6.D7.99.D7.94_.D7.A7.D7.93.D7.95.D7.9E.D7.94_.D7.91.D7.A7.D7.98.D7.A2_.D7.95.D7.A0.D7.95.D7.A1.D7.97.D7.AA_.D7.A0.D7.99.D7.95.D7.98.D7.95.D7.9F-.D7.9C.D7.99.D7.99.D7.91.D7.A0.D7.99.D7.A5"></span>קיום פונקציה קדומה בקטע ונוסחת ניוטון-לייבניץ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=7" title="עריכת קוד המקור של הפרק: קיום פונקציה קדומה בקטע ונוסחת ניוטון-לייבניץ"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=7" title="עריכת פסקה: "קיום פונקציה קדומה בקטע ונוסחת ניוטון-לייבניץ"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הוכחת המשפט במקרה פרטי של פונקציה רציפה. </p><p>אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> רציפה בכל הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c40bbeaa2f59e60b6259cebe2479bc24396f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.136ex; height:2.843ex;" alt="{\displaystyle \ [a,b]}"></span> אז היא בפרט אינטגרבילית בו (רציפות גוררת אינטגרביליות) ואז כפי שראינו קודם, הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f41f5bd8787016205747827b2bf6624bd4f44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.029ex; height:5.843ex;" alt="{\displaystyle \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span> מקיימת לכל נקודה שבה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> רציפה (במקרה זה, כל הקטע) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F'(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F'(x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57fa0b41636b778b7cbe49d509d71a083d19cfe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.735ex; height:3.009ex;" alt="{\displaystyle \ F'(x)=f(x)}"></span>. לכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98cf176d608f36193dd3bb15b4501f367550e940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.46ex; height:2.843ex;" alt="{\displaystyle \ F(x)}"></span> היא פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span> בקטע. </p><p>על פי הגדרה: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(b)-F(a)=\int _{a}^{b}f(t)\,\mathrm {d} t-\int _{a}^{a}f(t)\,\mathrm {d} t=\int _{a}^{b}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(b)-F(a)=\int _{a}^{b}f(t)\,\mathrm {d} t-\int _{a}^{a}f(t)\,\mathrm {d} t=\int _{a}^{b}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e8bca5a05f22ea180e7431fc643855a3bec6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.657ex; height:6.343ex;" alt="{\displaystyle \ F(b)-F(a)=\int _{a}^{b}f(t)\,\mathrm {d} t-\int _{a}^{a}f(t)\,\mathrm {d} t=\int _{a}^{b}f(t)\,\mathrm {d} t}"></span>. </p><p>כעת, כל שתי פונקציות קדומות של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span> נבדלות ביניהן בקבוע. כי נניח ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x),G(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x),G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844b26fa0c4cf91993a0529b00efb2b6101db64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.46ex; height:2.843ex;" alt="{\displaystyle \ F(x),G(x)}"></span> שתיהן פונקציות קדומות של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span>, אז <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left(F(x)-G(x)\right)'=f(x)-f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left(F(x)-G(x)\right)'=f(x)-f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773ecb6a90607a24d5833db996a071f845ff598d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.795ex; height:3.176ex;" alt="{\displaystyle \ \left(F(x)-G(x)\right)'=f(x)-f(x)=0}"></span>, כלומר הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x)-G(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x)-G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed02674d7ef8c4a828ca5217bd7ba8e443f9d064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.266ex; height:2.843ex;" alt="{\displaystyle \ F(x)-G(x)}"></span> היא קבוע, כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F(x)=G(x)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F(x)=G(x)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135ac9a7249931910004a21d6d64e8caec8196ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.372ex; height:2.843ex;" alt="{\displaystyle \ F(x)=G(x)+c}"></span>. </p><p>על כן: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ G(b)-G(a)=F(b)-c-\left(F(a)-c\right)=F(b)-F(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>G</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ G(b)-G(a)=F(b)-c-\left(F(a)-c\right)=F(b)-F(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a57b021f2cbedadea93c63058a1bc37a6717fcc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.957ex; height:2.843ex;" alt="{\displaystyle \ G(b)-G(a)=F(b)-c-\left(F(a)-c\right)=F(b)-F(a)}"></span>, וזאת <b>לכל</b> פונקציה קדומה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ G(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffccff18ee1e34e948d3d1564525bf54446f16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.546ex; height:2.843ex;" alt="{\displaystyle \ G(x)}"></span> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span>. </p><p>בזאת הושלמה הוכחת הנוסחה היסודית עבור פונקציה רציפה. </p><p><b>מש"ל</b> </p><p>הערות: </p> <ol><li>כדי להוכיח את המשפט עבור המקרה הכללי שf אינטגרבילית וF היא פונקציה קדומה של f בכל הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c40bbeaa2f59e60b6259cebe2479bc24396f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.136ex; height:2.843ex;" alt="{\displaystyle \ [a,b]}"></span>יש להשתמש בסכומי רימן ו<a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז'">משפט הערך הממוצע של לגראנז'</a>.</li> <li>המשפט אף נכון למקרה מוכלל בו f אינטגרבילית ופרט למספר סופי של נקודות F היא פונקציה קדומה של f ורציפה. ההוכחה דומה להערה 1.</li></ol> <div class="mw-heading mw-heading3"><h3 id="הוכחה_לנוסחת_ניוטון-לייבניץ_שאינה_מתבססת_על_המשפט_היסודי"><span id=".D7.94.D7.95.D7.9B.D7.97.D7.94_.D7.9C.D7.A0.D7.95.D7.A1.D7.97.D7.AA_.D7.A0.D7.99.D7.95.D7.98.D7.95.D7.9F-.D7.9C.D7.99.D7.99.D7.91.D7.A0.D7.99.D7.A5_.D7.A9.D7.90.D7.99.D7.A0.D7.94_.D7.9E.D7.AA.D7.91.D7.A1.D7.A1.D7.AA_.D7.A2.D7.9C_.D7.94.D7.9E.D7.A9.D7.A4.D7.98_.D7.94.D7.99.D7.A1.D7.95.D7.93.D7.99"></span>הוכחה לנוסחת ניוטון-לייבניץ שאינה מתבססת על המשפט היסודי</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=8" title="עריכת קוד המקור של הפרק: הוכחה לנוסחת ניוטון-לייבניץ שאינה מתבססת על המשפט היסודי"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=8" title="עריכת פסקה: "הוכחה לנוסחת ניוטון-לייבניץ שאינה מתבססת על המשפט היסודי"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(a=t_{0}<\cdots <t_{n}=b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(a=t_{0}<\cdots <t_{n}=b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f3d3130b4f1722ea112866d70e982c4e54886e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.95ex; height:2.843ex;" alt="{\displaystyle P=(a=t_{0}<\cdots <t_{n}=b)}"></span> חלוקה כלשהי של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c40bbeaa2f59e60b6259cebe2479bc24396f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.136ex; height:2.843ex;" alt="{\displaystyle \ [a,b]}"></span>. אז לפי <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז'">משפט הערך הממוצע של לגראנז'</a> מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x_{i}\in (t_{i-1},t_{i}):F(t_{i})-F(t_{i-1})=F'(x_{i})(t_{i}-t_{i-1})=f(x_{i})(t_{i}-t_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x_{i}\in (t_{i-1},t_{i}):F(t_{i})-F(t_{i-1})=F'(x_{i})(t_{i}-t_{i-1})=f(x_{i})(t_{i}-t_{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260505645e584ab6d2effe3b18bc85f74a83b111" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.652ex; height:3.009ex;" alt="{\displaystyle \exists x_{i}\in (t_{i-1},t_{i}):F(t_{i})-F(t_{i-1})=F'(x_{i})(t_{i}-t_{i-1})=f(x_{i})(t_{i}-t_{i-1})}"></span> עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ i=1,...,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ i=1,...,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f18aa9486b7ebab9fb6c9e5b91aa66d4aa26429a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.208ex; height:2.509ex;" alt="{\displaystyle \ i=1,...,n}"></span>; כעת נגדיר <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}={\underset {x\in [t_{i-1},t_{i}]}{\sup }}f(x);\quad m_{i}={\underset {x\in [t_{i-1},t_{i}]}{\inf }}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo form="prefix">sup</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </munder> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo form="prefix">inf</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </munder> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}={\underset {x\in [t_{i-1},t_{i}]}{\sup }}f(x);\quad m_{i}={\underset {x\in [t_{i-1},t_{i}]}{\inf }}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abba5dbc6928cad79705881dc1c4f885d08b0fec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.385ex; height:5.009ex;" alt="{\displaystyle M_{i}={\underset {x\in [t_{i-1},t_{i}]}{\sup }}f(x);\quad m_{i}={\underset {x\in [t_{i-1},t_{i}]}{\inf }}f(x)}"></span> ונקבל <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(t_{i}-t_{i-1})\leq f(x_{i})(t_{i}-t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}(t_{i}-t_{i-1})\leq f(x_{i})(t_{i}-t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2bf7959df52911cb3d9acb3c4dfc68fd8aef41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.394ex; height:2.843ex;" alt="{\displaystyle m_{i}(t_{i}-t_{i-1})\leq f(x_{i})(t_{i}-t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}"></span> כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(t_{i}-t_{i-1})\leq F(t_{i})-F(t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}(t_{i}-t_{i-1})\leq F(t_{i})-F(t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60d54b54edb184d11b0c740e28b4e84c845e8808" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.468ex; height:2.843ex;" alt="{\displaystyle m_{i}(t_{i}-t_{i-1})\leq F(t_{i})-F(t_{i-1})\leq M_{i}(t_{i}-t_{i-1})}"></span> נסכום את המשוואה האחרונה עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ i=1,...,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ i=1,...,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f18aa9486b7ebab9fb6c9e5b91aa66d4aa26429a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.208ex; height:2.509ex;" alt="{\displaystyle \ i=1,...,n}"></span> ונקבל: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(P)=\sum _{i=1}^{n}m_{i}(t_{i}-t_{i-1})\leq \sum _{i=1}^{n}F(t_{i})-F(t_{i-1})\leq \sum _{i=1}^{n}M_{i}(t_{i}-t_{i-1})=S(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(P)=\sum _{i=1}^{n}m_{i}(t_{i}-t_{i-1})\leq \sum _{i=1}^{n}F(t_{i})-F(t_{i-1})\leq \sum _{i=1}^{n}M_{i}(t_{i}-t_{i-1})=S(P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d4702cb0efff41ea54bcbfa99db9cdc04a9b86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.59ex; height:6.843ex;" alt="{\displaystyle s(P)=\sum _{i=1}^{n}m_{i}(t_{i}-t_{i-1})\leq \sum _{i=1}^{n}F(t_{i})-F(t_{i-1})\leq \sum _{i=1}^{n}M_{i}(t_{i}-t_{i-1})=S(P)}"></span> נשים לב ש- <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sum _{i=1}^{n}}F(t_{i})-F(t_{i-1})=F(b)-F(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sum _{i=1}^{n}}F(t_{i})-F(t_{i-1})=F(b)-F(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36cdadaf5786b3a09d270a908033ac7c9d22d97b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.941ex; height:6.843ex;" alt="{\displaystyle {\sum _{i=1}^{n}}F(t_{i})-F(t_{i-1})=F(b)-F(a)}"></span>(שהרי זהו <a href="/wiki/%D7%A1%D7%9B%D7%95%D7%9D_%D7%98%D7%9C%D7%A1%D7%A7%D7%95%D7%A4%D7%99" class="mw-redirect" title="סכום טלסקופי">סכום טלסקופי</a>) ונקבל: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(P)\leq F(b)-F(a)\leq S(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(P)\leq F(b)-F(a)\leq S(P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d2aacde713542bd97679af26ced92c4437057d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.064ex; height:2.843ex;" alt="{\displaystyle s(P)\leq F(b)-F(a)\leq S(P)}"></span> עבור <b>כל</b> חלוקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0699308adfbf0e364bf8a9b9efdb56720b33f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.326ex; height:2.176ex;" alt="{\displaystyle \ P}"></span>. אם כך <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C#הגדרה_באמצעות_סכומי_דארבו" title="אינטגרל">מהגדרת האינטגרל</a> התוצאה נובעת ישירות <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square \qquad \qquad \qquad F(b)-F(a)=\int _{a}^{b}f(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square \qquad \qquad \qquad F(b)-F(a)=\int _{a}^{b}f(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c37b7342558279750d5cb00b0d89800cc38f6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.226ex; height:6.343ex;" alt="{\displaystyle \square \qquad \qquad \qquad F(b)-F(a)=\int _{a}^{b}f(x)\,\mathrm {d} x}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="הכללות"><span id=".D7.94.D7.9B.D7.9C.D7.9C.D7.95.D7.AA"></span>הכללות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=9" title="עריכת קוד המקור של הפרק: הכללות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=9" title="עריכת פסקה: "הכללות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הכללה טבעית של המשפט היסודי של החדו"א לשני ממדים היא <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F" title="משפט גרין">משפט גרין</a>. בממדים גבוהים יותר קיימות הכללות מורכבות יותר, כגון <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%90%D7%95%D7%A1" title="משפט גאוס">משפט גאוס</a>, <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A1%D7%98%D7%95%D7%A7%D7%A1" title="משפט סטוקס">משפט סטוקס</a> ו<a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%A6%D7%99%D7%94_%D7%A9%D7%9C_%D7%9C%D7%91%D7%92&action=edit&redlink=1" class="new" title="משפט הדיפרנציאציה של לבג (הדף אינו קיים)">משפט הדיפרנציאציה של לבג</a> <small class="noprint" dir="rtl">(<a href="https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem" class="extiw" title="en:Lebesgue differentiation theorem">אנ'</a>)</small>. עבור אינטגרלים סופיים לא מחושבים, מתקיים <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A9%D7%99%D7%9E%D7%95%D7%A8_%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="משפט שימור האינטגרל">משפט שימור האינטגרל</a>. </p> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&action=edit&section=10" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99&veaction=edit&section=10" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterwikilinkT"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/" title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> מדיה וקבצים בנושא <b><a href="https://commons.wikimedia.org/wiki/Category:Fundamental_theorem_of_calculus" class="extiw" title="commons:Category:Fundamental theorem of calculus">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a></b> ב<a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A9%D7%99%D7%AA%D7%95%D7%A3" title="ויקישיתוף">וויקישיתוף</a></div> <ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/fundamental-theorem-of-calculus">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a>, באתר <a href="/wiki/%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="אנציקלופדיה בריטניקה">אנציקלופדיה בריטניקה</a> <span dir="rtl" class="languageicon">(באנגלית)</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36549940"></li></ul> <p>גדי אלכסנדרוביץ', <a rel="nofollow" class="external text" href="https://gadial.net/2011/01/02/fundemental_theorem_of_calculus/">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a>, באתר "לא מדויק", 2 בינואר 2011 </p><p><br /> </p> <table class="navbox nowraplinks mw-collapsible autocollapse" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA_-_%D7%9E%D7%95%D7%A0%D7%97%D7%99%D7%9D" title="אנליזה מתמטית - מונחים">חשבון אינפיניטסימלי</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה)">סדרה</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%9E%D7%AA%D7%9B%D7%A0%D7%A1%D7%AA" title="סדרה מתכנסת">סדרה מתכנסת</a> • <a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="גבול (מתמטיקה)">גבול</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A7%D7%95%D7%A9%D7%99" title="סדרת קושי">סדרת קושי</a> • <a href="/wiki/%D7%98%D7%95%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="טור (מתמטיקה)">טור</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C" title="אינפיניטסימל">אינפיניטסימל</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">שדה המספרים הממשיים</a> • <a href="/wiki/%D7%A2%D7%A8%D7%9A_%D7%9E%D7%95%D7%97%D7%9C%D7%98" title="ערך מוחלט">ערך מוחלט</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%94%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="אי-שוויון המשולש">אי-שוויון המשולש</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%A7%D7%95%D7%A9%D7%99-%D7%A9%D7%95%D7%95%D7%A8%D7%A5" title="אי-שוויון קושי-שוורץ">אי-שוויון קושי-שוורץ</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">פונקציות </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%9E%D7%A9%D7%99%D7%AA" title="פונקציה ממשית">פונקציה</a> • <a href="/wiki/%D7%92%D7%A8%D7%A3_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גרף של פונקציה">גרף פונקציה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="פונקציה ליניארית">פונקציה ליניארית</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%95%D7%A0%D7%95%D7%98%D7%95%D7%A0%D7%99%D7%AA" title="פונקציה מונוטונית">פונקציה מונוטונית</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F" title="נקודת קיצון">נקודת קיצון</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A4%D7%99%D7%AA%D7%95%D7%9C" title="נקודת פיתול">נקודת פיתול</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%95%D7%9B%D7%A3" title="נקודת אוכף">נקודת אוכף</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%A2%D7%95%D7%A8%D7%94" title="פונקציה קעורה">פונקציה קעורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%9E%D7%95%D7%A8%D7%94" title="פונקציה קמורה">פונקציה קמורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="פונקציה רציפה (אנליזה)">פונקציה רציפה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="פונקציה רציפה במידה שווה">פונקציה רציפה במידה שווה</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%99_%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" title="נקודת אי רציפות">נקודת אי רציפות</a> • <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> • <a href="/wiki/%D7%98%D7%95%D7%A8_%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A8" title="טור טיילור">טור טיילור</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA" title="סדרת פונקציות">סדרת פונקציות</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%A0%D7%A7%D7%95%D7%93%D7%AA%D7%99%D7%AA" title="התכנסות נקודתית">התכנסות נקודתית</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="התכנסות במידה שווה">התכנסות במידה שווה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">משפטים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%95%D7%9C%D7%A6%D7%90%D7%A0%D7%95-%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפט בולצאנו-ויירשטראס">משפט בולצאנו-ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפטי ויירשטראס">משפטי ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%A0%D7%98%D7%95%D7%A8_%D7%9C%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="משפט קנטור לרציפות במידה שווה">משפט קנטור</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A2%D7%A8%D7%9A_%D7%94%D7%91%D7%99%D7%A0%D7%99%D7%99%D7%9D" title="משפט ערך הביניים">משפט ערך הביניים</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A4%D7%A8%D7%9E%D7%94_(%D7%9C%D7%A0%D7%A7%D7%95%D7%93%D7%95%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F)" title="משפט פרמה (לנקודות קיצון)">משפט פרמה</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A8%D7%95%D7%9C" title="משפט רול">משפט רול</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז'">משפט הערך הממוצע של לגראנז'</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%A7%D7%95%D7%A9%D7%99" title="משפט הערך הממוצע של קושי">משפט הערך הממוצע של קושי</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%93%D7%90%D7%A8%D7%91%D7%95" title="משפט דארבו">משפט דארבו</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A9%D7%A8%D7%A9%D7%A8%D7%AA" title="כלל השרשרת">כלל השרשרת</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A1%D7%A0%D7%93%D7%95%D7%95%D7%99%D7%A5%27" title="כלל הסנדוויץ'">כלל הסנדוויץ'</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%9C%D7%95%D7%A4%D7%99%D7%98%D7%9C" title="כלל לופיטל">כלל לופיטל</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A9%D7%98%D7%95%D7%9C%D7%A5" title="משפט שטולץ">משפט שטולץ</a> • <a href="/wiki/%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%A9%D7%9C_%D7%92%D7%91%D7%95%D7%9C%D7%95%D7%AA" title="אריתמטיקה של גבולות">אריתמטיקה של גבולות</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">האינטגרל </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל">אינטגרל</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%90_%D7%90%D7%9E%D7%99%D7%AA%D7%99" title="אינטגרל לא אמיתי">אינטגרל לא אמיתי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%91-%D7%9E%D7%9E%D7%93%D7%99" title="אינטגרל רב-ממדי">אינטגרל רב-ממדי</a> • <a class="mw-selflink selflink">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94_%D7%91%D7%97%D7%9C%D7%A7%D7%99%D7%9D" title="אינטגרציה בחלקים">אינטגרציה בחלקים</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" title="שיטות אינטגרציה">שיטות אינטגרציה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">אנליזה מתקדמת </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="פונקציה מרוכבת">פונקציה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F-%D7%A8%D7%A4%D7%A1%D7%95%D7%9F" title="שיטת ניוטון-רפסון">שיטת ניוטון-רפסון</a> • <a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="משוואה דיפרנציאלית">משוואה דיפרנציאלית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> </td></tr> <tr> <td colspan="3" style="background-color: #F2F3F4; text-align: center; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית">אנליזה מתמטית</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת">אנליזה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99%D7%AA" title="אנליזה פונקציונלית">אנליזה פונקציונלית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> </td></tr> </tbody></table> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q1217677?uselang=he&#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q1217677?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q1217677?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" 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href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%9E%D7%A9%D7%A4%D7%98%D7%99%D7%9D_%D7%91%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94" title="קטגוריה:משפטים באנליזה">משפטים באנליזה</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99%D7%9D" title="קטגוריה:אינטגרלים">אינטגרלים</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%99%D7%99%D7%96%D7%A7_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F" title="קטגוריה:אייזק ניוטון">אייזק ניוטון</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">קטגוריות מוסתרות: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="קטגוריה:ערכים עם תבנית בריטניקה">ערכים עם תבנית בריטניקה</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_GND" title="קטגוריה:ויקיפדיה: ערכים עם מזהה GND">ויקיפדיה: ערכים עם מזהה GND</a></li></ul></div></div> </div> </main> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> דף זה נערך לאחרונה ב־21 ביולי 2024, בשעה 15:53.</li> <li id="footer-info-copyright">הטקסט מוגש בכפוף לרישיון <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.he">Creative Commons ייחוס-שיתוף זהה 4.0</a>; ייתכן שישנם תנאים נוספים. ר׳ את <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">תנאי השימוש</a> לפרטים.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a 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המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי – ויקיפדיה