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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="kmKT3AxT3YG1RuUDXRewOEL4ycQg4_T1Ar2d3qSmypdeTNPYssCTBYCtpUHk7lNhk2xvTBpVaOz1ex42nD12CA" /> <meta name="citation_title" content="Calculus for Economists Module teaching material" /> <meta name="citation_publication_date" content="2024/01/01" /> <meta name="citation_journal_title" content="Abdella Mohammed Ahmed(M.Sc.in Development economics)" /> <meta name="citation_author" content="Abdella M O H A M M E D Ahmed" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/125236892/Calculus_for_Economists_Module_teaching_material" /> <meta name="twitter:title" content="Calculus for Economists Module teaching material" /> <meta name="twitter:description" content=" This chapter is concerned with revising the basic concepts related with quantitative methods in economics. These concepts are discussed in detail in quantitative method for economists I&amp;II courses. So, the discussions in this chapter will be" /> <meta name="twitter:image" content="https://0.academia-photos.com/296672773/142781701/152558780/s200_abdella.ahmed.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/125236892/Calculus_for_Economists_Module_teaching_material" /> <meta property="og:title" content="Calculus for Economists Module teaching material" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content=" This chapter is concerned with revising the basic concepts related with quantitative methods in economics. These concepts are discussed in detail in quantitative method for economists I&amp;II courses. 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So, the discussions in this chapter will be" /> <title>(PDF) Calculus for Economists Module teaching material</title> <link rel="canonical" href="https://www.academia.edu/125236892/Calculus_for_Economists_Module_teaching_material" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'c766c41538e32ecdfb2b95f4f676348b47be9f37'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740011097000); window.Aedu.timeDifference = new Date().getTime() - 1740011097000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"This chapter is concerned with revising the basic concepts related with quantitative methods in economics. These concepts are discussed in detail in quantitative method for economists I\u0026amp;amp;II courses. So, the discussions in this chapter will be very brief. 1.1 The concept of calculus. Definition: calculus, in simple terms, in a mathematical technique which is used to deal with change, motion and growth. It is a result of great advancement in the area of mathematics which is used to solve mathematical problems that can’t be solved using ordinary methods. In economics, it is frequently used to solve static and dynamic economic problem. Differential calculus, which indicates the rate of change of the dependent variable with respect to the independent variable, is used in static economic analysis; On the other hand, Integral calculus, difference equation \u0026amp;amp; differential equation are use in dealing with dynamic economic problems. These applications are dealt with in a separate chapter but for this section you will be reminded about only the basic concepts which you have dealt with your Quantitative methods course. 1.1.1 Differential calculus 1. Derivatives Given a function the derivative of at any point ‘x’ is given by if this limit exists then is said to be differentiable function at ‘x’. The term is simply slope of the functions which is an equivalent expression to and it represents average rate of change of a function .But the derivative approximates the rate of change of the positive function when is infinitesimal i.e. slope of a function at a point or instantaneous rate of change. Example: find the derivative of Solution: Differentiability and continuity It the derivative of a function, say exists at point ‘c’ then is said to be differentiable at ‘c’. To say a function is differentiable at ‘c’ the following conditions should be fulfilled 1. should be continuous at ‘c’ 2. should have a unique (only one ) slope at ‘c’ The 1st condition requires the fulfillment of continuity conditions i.e. 1. should exist 2. 3. From this we can understand that continuity doesn’t necessarily mean differentiability. Continuality rules out jumps and gaps at a point while differentiability condition requires smoothness of the curve at the given point in addition to continuity conditions. In other words, differentiability rules out jumps, gaps, sharps \u0026amp;amp; cusp corners, while continuity rules out only the first two Example show the function is not differentiable at x = 2. Steps 1 Testing continuity of the function 1. 2. 3. so, is continuous at Step 2 in this step we will see weather the limit of the difference quotient exist or not. Since is an absolute value functions, it can be redefined as follows: To see the existence of the limit of the difference quotient, we find its left \u0026amp;amp; right side limits .i.e. From the above result, we can see that the left \u0026amp;amp; right side limits are not equal. There fore the limit of the difference quotient doesn’t exist which intern means that the function is not differentiable at x=2.","author":[{"@context":"https://schema.org","@type":"Person","name":"Abdella M O H A M M E D Ahmed","url":"https://independent.academia.edu/AbdellaMohammed39"}],"contributor":[],"dateCreated":"2024-11-02","dateModified":"2024-11-02","datePublished":"2024-01-01","headline":"Calculus for Economists Module teaching material","image":"https://attachments.academia-assets.com/119316546/thumbnails/1.jpg","inLanguage":"en","keywords":["Calculus"],"publication":"Abdella Mohammed Ahmed(M.Sc.in Development economics)","publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}],"thumbnailUrl":"https://attachments.academia-assets.com/119316546/thumbnails/1.jpg","url":"https://www.academia.edu/125236892/Calculus_for_Economists_Module_teaching_material"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: 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{"work":{"id":125236892,"created_at":"2024-11-02T07:35:55.191-07:00","from_world_paper_id":null,"updated_at":"2024-11-20T18:31:02.238-08:00","_data":{"abstract":" This chapter is concerned with revising the basic concepts related with quantitative methods in economics. These concepts are discussed in detail in quantitative method for economists I\u0026II courses. So, the discussions in this chapter will be very brief.\n1.1 The concept of calculus.\nDefinition: calculus, in simple terms, in a mathematical technique which is used to deal with change, motion and growth. It is a result of great advancement in the area of mathematics which is used to solve mathematical problems that can’t be solved using ordinary methods. In economics, it is frequently used to solve static and dynamic economic problem. Differential calculus, which indicates the rate of change of the dependent variable with respect to the independent variable, is used in static economic analysis; On the other hand, Integral calculus, difference equation \u0026 differential equation are use in dealing with dynamic economic problems. These applications are dealt with in a separate chapter but for this section you will be reminded about only the basic concepts which you have dealt with your Quantitative methods course. \n1.1.1 Differential calculus\n1.\tDerivatives\nGiven a function the derivative of at any point ‘x’ is given by \n if this limit exists then is said to be differentiable function at ‘x’.\nThe term is simply slope of the functions which is an equivalent expression to and it represents average rate of change of a function .But the derivative approximates the rate of change of the positive function when is infinitesimal i.e. slope of a function at a point or instantaneous rate of change.\nExample: find the derivative of \nSolution: \n\t \n \nDifferentiability and continuity\nIt the derivative of a function, say exists at point ‘c’ then is said to be differentiable at ‘c’.\n\tTo say a function is differentiable at ‘c’ the following conditions should be fulfilled\n\t1. should be continuous at ‘c’\n\t2. should have a unique (only one ) slope at ‘c’ \nThe 1st condition requires the fulfillment of continuity conditions i.e. \n1. should exist \n2. \n3. \nFrom this we can understand that continuity doesn’t necessarily mean differentiability.\nContinuality rules out jumps and gaps at a point while differentiability condition requires smoothness of the curve at the given point in addition to continuity conditions. In other words, differentiability rules out jumps, gaps, sharps \u0026 cusp corners, while continuity rules out only the first two \nExample show the function is not differentiable at x = 2.\t\nSteps 1 Testing continuity of the function\n 1. \n 2. \n 3. so, is continuous at \nStep 2 in this step we will see weather the limit of the difference quotient exist or not.\nSince is an absolute value functions, it can be redefined as follows:\n \nTo see the existence of the limit of the difference quotient, we find its left \u0026 right side limits .i.e.\n From the above result, we can see that the left \u0026 right side limits are not equal. There fore the limit of the difference quotient doesn’t exist which intern means that the function is not differentiable at x=2.\n","publication_date":"2024,,","publication_name":"Abdella Mohammed Ahmed(M.Sc.in Development economics)"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Calculus for Economists Module teaching material","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [296672773]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card ds-work-card--no-bottom-spacing"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":119316546,"attachmentType":"pdf"}"><img alt="First page of “Calculus for Economists Module teaching material”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/119316546/mini_magick20241102-1-cwdr8v.png?1730558137" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Calculus for Economists Module teaching material</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="296672773" href="https://independent.academia.edu/AbdellaMohammed39"><img alt="Profile image of Abdella M O H A M M E D Ahmed" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/296672773/142781701/152558780/s65_abdella.ahmed.jpg" />Abdella M O H A M M E D Ahmed</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2024, Abdella Mohammed Ahmed(M.Sc.in Development economics)</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">122 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 125236892; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=125236892"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md"> This chapter is concerned with revising the basic concepts related with quantitative methods in economics. These concepts are discussed in detail in quantitative method for economists I&II courses. So, the discussions in this chapter will be very brief. 1.1 The concept of calculus. Definition: calculus, in simple terms, in a mathematical technique which is used to deal with change, motion and growth. It is a result of great advancement in the area of mathematics which is used to solve mathematical problems that can’t be solved using ordinary methods. In economics, it is frequently used to solve static and dynamic economic problem. Differential calculus, which indicates the rate of change of the dependent variable with respect to the independent variable, is used in static economic analysis; On the other hand, Integral calculus, difference equation & differential equation are use in dealing with dynamic economic problems. These applications are dealt with in a separate chapter but for this section you will be reminded about only the basic concepts which you have dealt with your Quantitative methods course. 1.1.1 Differential calculus 1. Derivatives Given a function the derivative of at any point ‘x’ is given by if this limit exists then is said to be differentiable function at ‘x’. The term is simply slope of the functions which is an equivalent expression to and it represents average rate of change of a function .But the derivative approximates the rate of change of the positive function when is infinitesimal i.e. slope of a function at a point or instantaneous rate of change. Example: find the derivative of Solution: Differentiability and continuity It the derivative of a function, say exists at point ‘c’ then is said to be differentiable at ‘c’. To say a function is differentiable at ‘c’ the following conditions should be fulfilled 1. should be continuous at ‘c’ 2. should have a unique (only one ) slope at ‘c’ The 1st condition requires the fulfillment of continuity conditions i.e. 1. should exist 2. 3. From this we can understand that continuity doesn’t necessarily mean differentiability. Continuality rules out jumps and gaps at a point while differentiability condition requires smoothness of the curve at the given point in addition to continuity conditions. In other words, differentiability rules out jumps, gaps, sharps & cusp corners, while continuity rules out only the first two Example show the function is not differentiable at x = 2. Steps 1 Testing continuity of the function 1. 2. 3. so, is continuous at Step 2 in this step we will see weather the limit of the difference quotient exist or not. Since is an absolute value functions, it can be redefined as follows: To see the existence of the limit of the difference quotient, we find its left & right side limits .i.e. From the above result, we can see that the left & right side limits are not equal. 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