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Adds core trigonometric identities and BC-only topics such as integration by parts and arc length.",{"@graph":14,"@context":64},[15,34,55],{"@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/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/paper-templates/","Paper 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includes the First Derivative Test and Second Derivative Test for determining relative minima or maxima at critical numbers where f′(c)=0.","Answer","https://schema.org",{"og:url":32,"og:type":66,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":68,"canonical":32},"index,follow",{"doc_id":70,"site_id":7},573237,1792053097,{"code":4,"msg":73,"data":74},"success",[75,80,85,90,95,100,105,110,114],{"id":76,"doc_module":22,"doc_module_name":25,"category_name":77,"show_sort_weight":78,"slug":79},11,"Presentations",90,"presentations",{"id":81,"doc_module":22,"doc_module_name":25,"category_name":82,"show_sort_weight":83,"slug":84},12,"Resumes",80,"resumes",{"id":86,"doc_module":22,"doc_module_name":25,"category_name":87,"show_sort_weight":88,"slug":89},14,"Invoices",70,"invoices",{"id":91,"doc_module":22,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},15,"Posters",60,"posters",{"id":96,"doc_module":22,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},16,"Social 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CALCULUS FORMULA LIST  \n􀀁  1  􀀂 n  \nDefinition of e: e = ln 􀀅􀀃 1+ n 􀀆􀀄  \n\n| 􀀇x\u003Cbr>Absolute Value: x  = 􀀈\u003Cbr>􀀉−x |  |  | if\u003Cbr>if | x ≥ 0\u003Cbr>x \u003C 0 |  |  |\n| --- | --- | --- | --- | --- | --- | --- |\n| Definition of the Derivative: |  |  |  |  |  |  |\n| f ' (x )\u003Cbr>f ' (a )\u003Cbr>f ' (x ) | f (x + ∆x )− f (x )\u003Cbr>= lim ~~ ~~\u003Cbr>∆x→∞ ∆xf (a + h ) − f (a )\u003Cbr>= lim ~~  ~~h→∞ h\u003Cbr>f (x )− f (a )\u003Cbr>= lim ~~  ~~x→a x − a |  |  |  | f ' (x ) = lim  f (x + h ) − f (x )\u003Cbr>h→∞ h\u003Cbr>derivative at x = a\u003Cbr>alternate form |  |\n| Definition of Continuity:\u003Cbr>f is continuous at c iff: |  |  |  |  |  |  |\n| (1) f (c )\u003Cbr>(2) lim fx→c\u003Cbr>(3) lim fx→c |  | is defined\u003Cbr>(x ) exists\u003Cbr>(x ) = f (c ) |  |  |  |  |\n| Average Rate of Change of f (x ) on [a, b ] = ~~ ~~f~~ ~~(bb)~~ ~~−fa~~ ~~(b~~ ~~) |  |  |  |  |  |  |\n| Rolle's Theorem:\u003Cbr>If f is continuous on [a, b ] and differentiable on (a, b ) and if then there exists a number c on (a, b ) such that f ' (c ) = 0. |  |  |  |  |  | f (a ) = f (b ) , |\n\nMean Value Theorem:  \nIf f is continuous on [a, b ] and differentiable on (a, b ) , then there  \nexists a number c on (a, b ) such that f ' (c ) = ~~ ~~f~~ ~~(bb)~~ ~~−~~ ~~fa~~ ~~(a~~ ~~) .  \nNote : Rolle's Theorem is a special case of The Mean Value Theorem  \nIf f (a ) = f (b ) then f '(c ) = f~~ ~~(ab)−fa~~ ~~(b~~ ~~) = f~~ ~~(ab)−fa~~ ~~(a~~ ~~) = 0.  \nSource: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005  \nAP Calculus Formula List Math by Mr. Mueller Page 1 of 6  \nIntermediate Value Theorem:  \nIf f is continuous on [a, b ] and k is any number between f (a ) and f (b ) , then there is at least one number c between a and b such that f (c ) = k.  \n\n| Definition of a Critical Number:\u003Cbr>Let f be defined at c. If f ' (c ) = 0 or f ' is is undefined at c, then c is a critical number of f . |\n| --- |\n| First Derivative Test:\u003Cbr>Let c be a critical number of the function f that is continuous on an open interval I containing c. If f is differentiable on I , except possibly at c, then f (c ) can be classified as follows.\u003Cbr>(1) If f ' (x ) changes from negative to positive at c, then f (c ) is a relative minimum of f .\u003Cbr>(2 ) If f ' (x ) changes from positive to negative at c, then f (c ) is a relative maximum of f . |\n| Second Derivative Test:\u003Cbr>Let f be a function such that f ' (c ) = 0 and the second derivative existson an open interval containing c.\u003Cbr>(1) If f \" (c ) > 0, then f (c ) is a relative minimum.\u003Cbr>(2 ) If f \" (c ) \u003C 0, then f (c ) is a relative maximum. |\n| Definition of Concavity:\u003Cbr>Let f be differentiable on an open interval I. The graph of f is concave upward on I if f 'is increasing on the interval, and concave downward on I , if f ' is decreasing on the interval. |\n| Test for Concavity:\u003Cbr>Let f be a function whose second derivative exists on an open interval I.\u003Cbr>(1) If f \" (x ) > 0 for all x in I , then the graph of f is concave upward on I.\u003Cbr>(2 ) If f \" (x ) \u003C 0 for all x in I , then the graph of f is concave downward on I. |\n| Definition of an Inflection Point:\u003Cbr>A function f has an inflection point at ( c, f (c )) if\u003Cbr>(1) f \" (c ) = 0 or f \" (c ) does not exist, and if\u003Cbr>(2 ) f changes concavity at x = c. |\n\nExponential Growth:  \ndydt = ky y (t) = Cekt  \nSource: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005  \nAP Calculus Formula List Math by Mr. Mueller Page 2 of 6  \nDerivative of an Inverse Function: ( f −1) ' (x ) = ~~ ~~f~~ ~~'~~ ~~(~~ ~~f~~ ~~1~~ ~~(x~~ ~~)~~ ~~)  \n\n| First Fundamental Theorem of Calculus:\u003Cbr>∫ab f ' (x ) dx = f (b ) − f (a ) |\n| --- |\n| Second Fundamental Theorem of Calculus:\u003Cbr>ddx~~ ~~ ∫ ax f (t ) dt = f (x )\u003Cbr>ddx~~ ~~ ∫ ga(x ) f (t ) dt = f ( g (x ))⋅ g ' (x ) Chain Rule Version |\n| The Average Value of a Function: |\n| Average value of f (x ) on [a, b ] fAVE = b~~ ~~~~ ~~a~~ ~~ ∫baf (x ) dx |\n| Volume of Revolution:\u003Cbr>Volume around a horizontal axis by discs: V = π ∫ba r","cbCaiobgj4siFmwe","https://ap.wps.com/l/cbCaiobgj4siFmwe","pdf",68683,6,"English","# AP Calculus Formula List\n## Definitions and Core Theorems\n## Derivative and Continuity Tests\n## Applications and Related Models\n## Trigonometric Identities\n## Calculus BC Only","[{\"question\":\"What tests are listed for classifying critical points?\",\"answer\":\"The formula list includes the First Derivative Test and Second Derivative Test for determining relative minima or maxima at critical numbers where f′(c)=0.\"}]","AP Calculus Formula List - reference | PDF",1791935440]