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Covers definite integrals, the fundamental theorems of calculus, optimization tests, concavity and inflection point criteria, and applications including L’Hospital’s Rule, average/instantaneous rates, and particle motion relationships.",{"@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/ap-calculus-bc-formula-list/311539/",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/ap-calculus-bc-formula-list/311539.png","ImageObject",300,407,{"name":42,"@type":43},"Finn","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-20",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",22,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What conditions define continuity at a point c?","Question",{"text":63,"@type":64},"Continuity at c requires that f(c) is defined, the limit limx→c f(x) exists, and limx→c f(x) equals f(c).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do the Mean Value Theorem and Intermediate Value Theorem differ?",{"text":68,"@type":64},"The Mean Value Theorem guarantees a point c where f'(c) equals the average rate of change on (a, b) under continuity and differentiability. The Intermediate Value Theorem guarantees that for any value k between f(a) and f(b), there exists c in [a, b] with f(c)=k under continuity.",{"name":70,"@type":61,"acceptedAnswer":71},"Which tests help determine local extrema and concavity?",{"text":72,"@type":64},"First and second derivative tests classify relative minima and maxima using sign changes in f'(x) or f''(x). Concavity and inflection points are determined using the sign of f''(x) and whether f''(x) changes sign (or the function transitions between increasing/decreasing).","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},311539,1789894429,{"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":96,"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},34359740700684,"https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487","2020 AP CALCULUS BC FORMULA LIST  \nDefinition of the derivative:  \n􀁣 f 􀀋x 􀀎 h􀀌􀀐 f 􀀋x􀀌 􀁣 f 􀀋x􀀌􀀐 f 􀀋a􀀌  \nf (x) 􀀠 lhim􀁯0  h  f 􀀋a􀀌 􀀠 lxim􀁯a  x 􀀐 a  (Alternative form)  \nDefinition of continuity: f is continuous at c if and only if  \n1) f (c) is defined;  \n2) limx􀁯c f (x) exists;  \n3) limx􀁯c f (x) 􀀠 f (c) .  \nMean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c on (a, b) such that f 􀁣 (c) 􀀠 f~~ ~~􀀋b􀀌b~~ ~~􀀐~~ ~~fa~~ ~~􀀋a􀀌 .  \nIntermediate Value Theorem: If f is continuous on [a, b] and k is any number between f (a)  \nand f (b), then there is at least one number c between a and b such that f (c) = k.  \nddx xn  􀀠 nxn􀀐1 ~~ ~~ddx u~~ ~~  􀀠 d2u~~dx~~u ddx  f 􀀋x 􀀌g 􀀋x 􀀌  􀀠 f 􀀋x 􀀌g 􀁣 􀀋x 􀀌􀀎 g 􀀋x 􀀌 f 􀁣 􀀋x 􀀌 ddx 􀀋 f 􀀋 g 􀀋x 􀀌􀀌􀀌 􀀠 f 􀁣 􀀋 g 􀀋x 􀀌􀀌􀂘 g 􀁣 􀀋x 􀀌  \nddx [sin u] 􀀠 cosu ddux ddx [tan u] 􀀠 sec2 u ddux  \n d   du  \n[sec u] 􀀠 secu tanudx dx  \n d   1 du  \n[ln u] 􀀠 dx u dx  \nddx [eu ] 􀀠 eu ddux  \n d   1 du  \n[arcsin u] 􀀠 dx 1 􀀐 u  dx  \n d   1 du  \n[arctan u] 􀀠 2  \ndx 1 􀀎 u dx  \nd  􀂪  1  􀂺  1   \ndx 􀂬􀂫 x 􀂼􀂻 􀀠 􀀐 x2  \nd 􀂪 f 􀀋x 􀀌􀂺 g 􀀋x 􀀌 f 􀁣 􀀋x 􀀌􀀐 f 􀀋x 􀀌g 􀁣 􀀋x 􀀌 ~~ ~~ 􀂫 ~~ ~~ 􀂻 􀀠  2   \ndx 􀂬g 􀀋x 􀀌 􀂼 􀀋 g 􀀋x 􀀌􀀌  \nddx [cos u] 􀀠 􀀐 sin u ddux ddx [cot u] 􀀠 􀀐 csc2 u ddux  \n d   du  \n[csc u] 􀀠 􀀐 cscu cot udx dx  \n d   1 du  \n[log u] 􀀠 dx a ulna dx  \nddx [au ] 􀀠 au ln a ddux  \n d   1 du  \n[arccosu] 􀀠 􀀐 dx 1 􀀐 u  dx  \n d   1 du  \ndx [arc cot u] 􀀠 􀀐 1 􀀎 u 2 dx  \nDefinition of a definite integral: 􀂳ba f  \nn 􀀋x 􀀌dx 􀀠 l􀀧ixm􀁯0 1 f  \nn 􀀋xk 􀀌􀂘 􀀋􀀧xk 􀀌 􀀠 lnim􀁯􀁦 􀂦 f  \nk 􀀠 1  \n􀀋xk 􀀌􀂘 􀀋􀀧xk 􀀌  \n􀂳 cosu du 􀀠 sin u 􀀎 C  \n􀂳 xn dx 􀀠 xnn~~ ~~􀀎􀀎11 􀀎 C, n 􀁺 􀀐1  \n􀂳 eu du 􀀠 eu 􀀎 C  \n􀂳 f 􀁣 􀀋 g 􀀋x 􀀌􀀌􀂘 g 􀁣 􀀋x 􀀌dx 􀀠 f  \n􀀋 g 􀀋x 􀀌􀀌 􀀎 C  \n􀂳 sin u du 􀀠 􀀐 cosu 􀀎 C  \n􀂳 1u du 􀀠 ln u  􀀎 C  \n􀂳 au du 􀀠 lanua 􀀎 C  \nDefinition of a Critical Number:  \nLet f be defined at c. If f 􀁣 􀀋c􀀌 􀀠 0 or if f 􀁣 is undefined at c, then c is a critical number of f.  \nFirst Derivative Test:  \nLet c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f 􀀋c􀀌 can be classified:  \n1) If f 􀁣 􀀋x􀀌 changes from negative to positive at c, then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative minimum off.  \n2) If f 􀁣 􀀋x􀀌 changes from positive to negative at c, then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative  maximum off.  Second Derivative Test:  \nLet f be a function such that the second derivative of f exists on an open interval containing c.  \n1) If f 􀁣 􀀋c􀀌 􀀠 0 and f 􀁣􀁣 􀀋c􀀌 􀀡 0 , then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative minimum.  \n2) If f 􀁣 􀀋c􀀌 􀀠 0 and f 􀁣􀁣 􀀋c􀀌 􀀟 0 , then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative maximum  \nDefinition of Concavity:  \nLet 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.  \nTest for Concavity:  \nLet f be a function whose second derivative exists on an open interval I.  \n1) If f 􀁣􀁣 􀀋x􀀌 􀀡 0 for all x in I, then the graph of f is concave upward in I.  \n2) If f 􀁣􀁣 􀀋x􀀌 􀀟 0 for all x in I, then the graph of f is concave downward in I.  \nDefinition of an Inflection Point:  \nA function f has an inflection point at 􀀋 c, f 􀀋c􀀌􀀌  \n1) if f 􀁣􀁣 􀀋c􀀌 􀀠 0 or f 􀁣􀁣 􀀋c􀀌 does not exist and  \n2) if f 􀁣􀁣 changes sign from positive to negative or negative to positive at x 􀀠 c  \nOR if f 􀁣 􀀋x 􀀌 changes from increasing to decreasing or decreasing to increasing at x = c.  \nFirst Fundamental Theorem of Calculus: 􀂳ba f 􀁣 􀀋x􀀌dx 􀀠 f 􀀋b􀀌􀀐 f 􀀋a􀀌  \nf 􀀋b 􀀌 􀀠 f 􀀋a 􀀌 􀀎 􀂳 ba f 􀁣 􀀋x 􀀌dx  \nf 􀀋a 􀀌 􀀠 f 􀀋b 􀀌 􀀐 􀂳ba f 􀁣 􀀋x 􀀌dx  \nfinal 􀀠 initial + change  \ninitial 􀀠 final 􀀐 change  \nSecond Fundamental Theorem of Calculus: ddx 􀂳 xa f 􀀋t 􀀌dt 􀀠 f 􀀋x􀀌  \nChain Rule Version: ~~ ~~ddx 􀂳 ag 􀀋x􀀌 f 􀀋t 􀀌dt 􀀠 f 􀀋 g 􀀋x􀀌􀀌􀂘 g 􀁣 􀀋x􀀌  \nAverage rate of change of f (x) on [a, b]:   \n_____________________________________________________________________________________________  \nAverage value of f (x) on [a, b]: fAVE 􀀠 b~~ ~~~~ ~~a  \n􀂳ba f (x)dx  \nParticle Motion  \nIf an object moves along a straight line with position function s ","cbCaijubWAn8lB0D","https://ap.wps.com/l/cbCaijubWAn8lB0D","pdf",727553,"English","# Derivative and Limit Foundations\n## Continuity, MVT, and IVT\n## Differentiation Rules and Trig/Log Derivatives\n# Integral Foundations and Applications\n## Definite Integrals and Fundamental Theorems\n## Integration by Parts and Common Integral Forms\n## L’Hospital’s Rule\n# Optimization, Concavity, and Motion\n## Critical Numbers and Derivative Tests\n## Concavity and Inflection Points\n## Particle Motion and Rate In/Out","[{\"question\":\"What conditions define continuity at a point c?\",\"answer\":\"Continuity at c requires that f(c) is defined, the limit limx→c f(x) exists, and limx→c f(x) equals f(c).\"},{\"question\":\"How do the Mean Value Theorem and Intermediate Value Theorem differ?\",\"answer\":\"The Mean Value Theorem guarantees a point c where f'(c) equals the average rate of change on (a, b) under continuity and differentiability. The Intermediate Value Theorem guarantees that for any value k between f(a) and f(b), there exists c in [a, b] with f(c)=k under continuity.\"},{\"question\":\"Which tests help determine local extrema and concavity?\",\"answer\":\"First and second derivative tests classify relative minima and maxima using sign changes in f'(x) or f''(x). Concavity and inflection points are determined using the sign of f''(x) and whether f''(x) changes sign (or the function transitions between increasing/decreasing).\"}]","AP Calculus BC Formula List | PDF",13]