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Also summarizes applications such as exponential growth/decay, Newton’s method, implicit differentiation, related rates, maxima/minima, slope fields, and integration methods (basic integrals, substitution, long division, parts, trig substitution, numerical integration) with geometry applications.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/quick-review-sheet-for-ap-calculus-exam-limits-derivatives-applications-of-derivatives/311688/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/quick-review-sheet-for-ap-calculus-exam-limits-derivatives-applications-of-derivatives/311688.png","ImageObject",300,407,{"name":42,"@type":43},"McQueen","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-21","2026-09-20",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",18,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the main steps for evaluating limits on this review sheet?","Question",{"text":63,"@type":64},"Use substitution, then factor and simplify as needed. For rational expressions taking limits as x approaches infinity, divide by the highest power of x.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the sheet determine concavity and inflection points?",{"text":68,"@type":64},"Concavity up uses f''(x) > 0 and concavity down uses f''(x) \u003C 0. Inflection is indicated when concavity changes; a second derivative test uses sign change around points where the second derivative is zero, noting that f''(0)=0 alone does not guarantee an inflection point.",{"name":70,"@type":61,"acceptedAnswer":71},"Which derivative tests are provided for relative extrema?",{"text":72,"@type":64},"The sheet includes the first-derivative test (sign changes of f'(x) around critical points) and a second-derivative test that classifies maximum/minimum using the sign of f''(critical point), with a possible inflection when it is zero.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},311688,1789894649,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":124,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144,"preview_page_count":4},5909890329169,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Quick Review Sheet for A.P. Calculus Exam  \nName   \nAP Calculus AB/BC Date Period  \n\n| Limits\u003Cbr>\u003Cbr>1.  Definition:\u003Cbr>\u003Cbr>\u003Cbr>2. Steps in Evaluating Limits:\u003Cbr>- Substitute, Factor, and Simplify\u003Cbr>\u003Cbr>\u003Cbr>3. Limits as x approaches infinity\u003Cbr>If taking the limit of a Rational expression, Divide by the highest power of x.\u003Cbr>\u003Cbr>4. L'Hopital's Rule\u003Cbr>\u003Cbr>5. Trig Limits \u003Cbr>|\n| --- |\n| Continuity |\n\n1. Definition  \n| Derivatives\u003Cbr>1. Definition: |  |\n| --- | --- |\n| •\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>•\u003Cbr>• | |\n\n| •\u003Cbr>•\u003Cbr>•\u003Cbr>• | |\n| --- | --- |\n\nApplications of Derivatives  \nAnalysis of Functions  \n1. Increasing / Decreasing  \nIf f'(x) > 0 Then f(x) is increasing.  \nIf f'(x) \u003C 0 Then f(x) is decreasing.  \nIf f'(x) = 0 Then f(x) is constant.  \n2. Concavity  \nIf f''(x) > 0, then f(x) is concave up. (Mr. Happy Face)  \nIf f''(x) \u003C 0, then f(x) is concave down. (Mr. Frowny)  \nPoints of inflection occur when the concavity changes. Test: If there is a point of inflection, the second  \nderivative is zero.  \nBUT just because the second derivative is zero doesn't guarantee a point of inflection.  \n3. Relative Extrema  \n1st derivative test:  \nTest points on each side of the critical points found by substituting in the first derivative.  \nIf the value of the derivative of the point to the left of the critical point is positive and the value of the derivative for the point to the right is negative, then the critical point is a relative maximum.  \nIf the value of the derivative of the point to the left of the critical point is negative and the value of the derivative for the point to the right is positive, then the critical point is a relative minimum.  \nIf the values of the derivative of the points to the left and the right of the critical point are the same (i.e., both positive or both negative), then the critical point is a point of inflection.  \n2nd derivative test:  \nTake the first derivative and set it equal to zero to solve for critical points.  \nTake the second derivative of the function.  \nSubstitute the critical point in the second derivative.  \nIf this value is negative, the critical point is a relative maximum.  \nIf this value is positive, the critical point is a relative minimum.  \nIf this value is zero, the critical point is a possible point of inflection. Test points on either side of the critical point by substituting them into the second derivative to verify that the concavity changed.  \n4. Mean Value Theorem  \n5. Exponential Growth and Decay  \n• Newton's Method  \n• Implicit Differentiation  \n• Related Rates  \n• Applied Maxima / Minima  \n• Slope Fields  \n| Some Basic Indefinite Integrals\u003Cbr>• \u003Cbr>• \u003Cbr>• \u003Cbr>• ∫ sin x dx = -cos x + C\u003Cbr>• ∫ cos x dx = sin x + C\u003Cbr>• ∫ sec ² x dx = tan x + C\u003Cbr>• ∫ csc x cot x dx = - csc x + C\u003Cbr>• ∫ sec x tan x dx = sec x + C\u003Cbr>• ∫ csc ² x dx = - cot x + C • \u003Cbr>•  |\n| --- |\n|  |\n\n\n| Methods of Integration |\n| --- |\n| If you cannot simply integrate using the basic formulas above, try one of the following methods: |\n| 1.Integration by substitution: a method by which some variable is substituted for part of the function f(x) (generally some complicated function within the larger function) in order to make integration simpler.\u003Cbr>2. Integration using long division: If the integral involves a quotient in which the degree of the numerator is greater than or equal to the degree of the denominator, divide the numerator by the denominator.\u003Cbr>3. Integration by parts: a method of integrating two functions multiplied together (the opposite of the product rule for derivatives), following the formula:\u003Cbr>|\n\n4. Integration by Trig Substitution:  \nIf the integral contains trig expressions, try substituting some of the basic trig identities:  \nIf the integral contains the sum or difference of two squares, set up right triangles and make appropriate trig substitutions.  \n5. Numerical Integration:  \nWhen symbolic methods fail, use of some numerical approximation method will give useful ans","cbCaipQ0Kyy7L2sq","https://ap.wps.com/l/cbCaipQ0Kyy7L2sq","pdf",406948,"English","# Limits\n## Limits as x approaches infinity\n## L'Hôpital's Rule\n## Trig Limits\n# Continuity\n# Derivatives\n## Applications of Derivatives\n### Increasing / Decreasing\n### Concavity\n### Relative Extrema\n### Mean Value Theorem\n### Exponential Growth and Decay\n# Integration\n## Basic Indefinite Integrals\n## Methods of Integration\n## Applications of Integration\n# Miscellaneous Geometry and Algebra Formulas\n# Trigonometric Formulas\n# Some Geometry Figures","[{\"question\":\"What are the main steps for evaluating limits on this review sheet?\",\"answer\":\"Use substitution, then factor and simplify as needed. For rational expressions taking limits as x approaches infinity, divide by the highest power of x.\"},{\"question\":\"How does the sheet determine concavity and inflection points?\",\"answer\":\"Concavity up uses f''(x) \\u003e 0 and concavity down uses f''(x) \\u003c 0. Inflection is indicated when concavity changes; a second derivative test uses sign change around points where the second derivative is zero, noting that f''(0)=0 alone does not guarantee an inflection point.\"},{\"question\":\"Which derivative tests are provided for relative extrema?\",\"answer\":\"The sheet includes the first-derivative test (sign changes of f'(x) around critical points) and a second-derivative test that classifies maximum/minimum using the sign of f''(critical point), with a possible inflection when it is zero.\"}]","Quick Review Sheet for A.P. Calculus Exam - Limits - Derivatives - Applications of Derivatives | PDF",25]