[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-313937-105":59,"doc-detail-313937-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","bc-calc-memorization-sheet","BC Calc Memorization Sheet","","BC Calculus memorization sheet compiling core differentiation and integration rules, limits, and foundational theorems used in calculus problem solving. Includes derivative shortcuts for common functions (trig, exponential, logarithmic, inverse trig), product/quotient/chain rule formulas, and key limit and integral properties such as the Fundamental Theorems of Calculus, Riemann sums, and common integral results. Also covers series expansions, Taylor/Maclaurin series, convergence tests, and applications including average/instantaneous rates of change, mean value/continuity theorems, and arc length and polar coordinate formulas.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/bc-calc-memorization-sheet/313937/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/bc-calc-memorization-sheet/313937.png","ImageObject",300,407,{"name":92,"@type":93},"Eliana","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-05","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",21,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the main derivative rules summarized in this sheet?","Question",{"text":113,"@type":114},"It lists derivative shortcuts and core rules including product, quotient, and chain rule along with formulas for common functions such as sin, cos, tan, ln, and exponentials.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Which integral theorems and definite integral identities are included?",{"text":118,"@type":114},"The sheet includes the Fundamental Theorems of Calculus, along with start-plus-accumulation relations, Riemann sum guidance, and standard integral results for powers and basic trig/exponential/log forms.",{"name":120,"@type":111,"acceptedAnswer":121},"What series expansions and convergence tests are provided?",{"text":122,"@type":114},"It provides elementary series plus Taylor and Maclaurin series structure, and includes multiple convergence/divergence tests such as ratio/limit comparison and other standard criteria.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},313937,1789899646,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":24,"preview_page_count":4},4398048949847,"https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267","BC Calc Memorization Sheet  \n\n| Derivatives\u003Cbr>Prod. ddx (uv ) = uv 􀁣 + vu 􀁣 d t bt 􀁣 − tb􀁣\u003Cbr>Quot. ~~ ~~ ~~ ~~ = ~~  ~~dx b b2\u003Cbr>Chain ddx f (u ) = f 􀁣 (u ) u 􀁣 or ~~ ~~ddxf (g(x ))=f 􀁣(g(x ))g 􀁣(x ) ddx xn = nxn−1\u003Cbr> d  ln x =  1   d  ln u = u􀁣 dx x , dx u\u003Cbr>ddx loga x = x~~ ~~1lna\u003Cbr> d  ex = ex  d  ax = ax ln adx , dx\u003Cbr>ddx sin x = cos xddx cos x = − sin xddx tan x = sec2 xddx cotx = − csc2 xddx sec x = sec x tan x\u003Cbr>ddx cscx =− cscx cotxd 1\u003Cbr>~~ ~~ arcsin x = ~~    ~~dx 1 x \u003Cbr> d   1 \u003Cbr>dx arctan x = 1 + x2\u003Cbr>􀁣 f (x + 􀀧x) − f (x) f (x) = lim ~~ ~~\u003Cbr>􀀧x→0 􀀧x\u003Cbr>􀁣 f (x) − f (a) f (a) = lim ~~ ~~\u003Cbr>x→a x − a\u003Cbr>􀁣 f (a + h )− f (a ) f (a ) = lh0  h  | Integrals\u003Cbr>􀂳 xn dx = xn11 + C , n 􀁺 −1\u003Cbr>􀂳 ~~ ~~1x dx = ln x  + C\u003Cbr>􀂳 uu􀁣 dx = ln u  + C\u003Cbr>􀂳 ex dx = ex + C\u003Cbr>􀂳 ax dx = lanxa + C\u003Cbr>􀂳 cos x dx = sin x + C\u003Cbr>􀂳 sin x dx = −cos x + C\u003Cbr>􀂳 sec2 x dx = tan x + C\u003Cbr>􀂳 tan x dx = −lncos x + C\u003Cbr>􀂳 cotx dx = lnsin x + C\u003Cbr>􀂳 sec x tan x dx = sec x + C\u003Cbr>􀂳 csc2 x dx = −cotx + C\u003Cbr>􀂳 cscx cotx dx = −cscx + C\u003Cbr>􀂳 ~~ ~~a2~~ ~~􀁣~~ ~~u~~ ~~2~~ ~~ du = 1a arctan ~~ ~~ua + C\u003Cbr>􀂳 ~~ ~~~~ ~~a2u~~ ~~u~~ ~~2~~ ~~ du = arcsin ~~ ~~ua + C Parts 􀂳 udv = uv − 􀂳v du\u003Cbr>|  |  |  |\n| --- | --- | --- | --- | --- |\n|  | Logistic\u003Cbr>dPdt = kP ( M − P ) M = carrying capacity Fastest growth is when\u003Cbr>1 P = M .\u003Cbr>2 |  |  |  |\n|  |  |  |  |  |\n\nFundamental Theorem  \n􀂳ba f 􀁣 (x ) dx =f (b )− f (a )  \nStart plus Accumulation  \nf (b ) = f (a ) + 􀂳ba f 􀁣 (x ) dx  \nSecond Fundamental Theorem  \nddx 􀂳uv f (t ) dt = f (v )v 􀁣 − f (u )u 􀁣  \nArea of Trapezoid A = 12 w (h1+h2 )  \nRiemann Sum (add areas of rectangles)  \n\n| Alt. Series Error: error 􀁤 an+1 (the next term) |  |  |\n| --- | --- | --- |\n| Lagrange Error:\u003Cbr>error 􀁤\u003Cbr>f (n+1) (z ) (x − c )n+1 (n +1)!\u003Cbr>is a maximum between x\u003Cbr>where\u003Cbr>and c | f (n+1) (z ) |  |\n\nElementary Series  \nx2 x3 xn  \nex = 1+ x + ~~ ~~ + ~~ ~~ +􀂘􀂘􀂘+ ~~ ~~ +􀂘􀂘􀂘  \n2! 3! n!  \nsin x = x − x3  + x5  − 􀂘􀂘􀂘 + (−1)n x2n+1  + 􀂘􀂘􀂘 3! 5! (2n +1)! x2 x4 (−1)n x2n  \ncos x = 1 − ~~ ~~ + ~~ ~~ − 􀂘􀂘􀂘 + ~~ ~~ + 􀂘􀂘􀂘 2! 4! (2n )!  \nTaylor Series  \nf 􀁣􀁣 (c ) 2 f 􀁣􀁣􀁣 (c ) 3 f (n ) (c ) n  \nf (x ) = f (c ) + f 􀁣 (c )(x − c ) +  2! (x − c ) +  3!  (x − c ) + 􀂘􀂘􀂘+  n!  (x − c ) + 􀂘􀂘􀂘 Maclaurin Series ( Taylor series with c = 0 )  \nEuler’s  \n\n| x | y | dy\u003Cbr>dx | 􀀧y = m􀀧x |\n| --- | --- | --- | --- |\n|  |  |  |  |\n\nGary Taylor [taylormathconsulting.com](taylormathconsulting.com)  \nAverage Rate of Change: AROC = f (b) − f (a) (slope between two pts)  \nb − a  \nInst. Rate of Change: IROC = f 􀁣 (c ) (slope at a single point)  \nMean Value Thm.: f 􀁣 (c) = f~~ ~~(bb)~~ ~~−~~ ~~fa~~ ~~(a) (find c where msec = mtan) Must be cont. on 􀀾a, b 􀁀 and differentiable on (a, b )  \nAverage Value of a Function:  􀂳baf (x )dx 􀂧  area 􀂷  \nfavg = b − a 􀂩􀂨 avg. height = width 􀂹􀂸  \nIntermediate Value Thm.: A function f that is continuous on 􀀾a, b 􀁀 takes on every y-value between f (a ) and f (b ) . Extreme Value Thm.: A function f that is continuous on 􀀾a, b 􀁀 has both an abs. min. and an abs. max. on the interval. Definition of Continuity: A function f is continuous at an x-value c if and only if lxc − f (x ) = lxc+ f (x ) = f (c ) .  \nSqueeze Theorem:~~ ~~If f (x ) 􀁤 g (x ) 􀁤 h(x ) for all x 􀁺 c in some interval containing c and if c f (x ) = L = c h (x ) , then c g (x ) = L.  \n\n| Arc Length = 􀂳ba | \u003Cbr> ( ( ))\u003Cbr>\u003Cbr>2\u003Cbr>1+ f 􀁣 x | dx = 􀂳tt12 |  ( ~~ ~~dxdt )2 + ( ~~ ~~dydt )2  | dt | |\n| --- | --- | --- | --- | --- | --- |\n|  |  | parametric, vectors |  |  |  |\n| Speed = v (t ) = ( ~~ ~~dxdt )2 + ( ~~ ~~dydt )2 Speed is inc. when Total Dist. = 􀂳tt12 v (t ) dt = 􀂳tt12\u003Cbr>vel. and acc. have\u003Cbr>vectors\u003Cbr>Polar Area = 12 􀂳􀁔􀁔12 r 2 d􀁔 Pathe same signrametric Der.ivatives: dydx = ~~  ~~dy~~dt~~dxdt~~  ~~ , d2~~ ~~ydx2 = ~~  ~~ddt~~ ~~􀂧􀂩dd􀂨~~ ~~ddxtyx􀂷􀂹􀂸\u003Cbr>\u003Cbr>( ~~ ~~dxdt )2 + ( ~~ ~~dydt )2 dt vectors\u003Cbr>\u003Cbr>\u003Cbr>Polar Conversions: r 2 = x2 + y2 , x = r cos􀁔 , y = r sin􀁔, 􀁔 = arctan yx |  |  |  |  |  |\n\nTests for Convergence/Divergence  \n\n| nth term test | div. ","cbCaiqa2NqU3JOLy","https://ap.wps.com/l/cbCaiqa2NqU3JOLy","pdf",119413,"English","# Derivatives\n## Rules and common derivatives\n## Limits and theorems\n# Integrals\n## Fundamental Theorems of Calculus\n## Common integrals and Riemann sums\n# Series and Approximations\n## Elementary series\n## Taylor and Maclaurin series\n## Error bounds\n# Average/Instantaneous Rates and Continuity\n## Mean Value and Extreme Value Theorems\n## Squeeze theorem and continuity\n# Applications\n## Arc length, parametric curves, polar coordinates\n# Series Convergence Tests\n## Comparison, limit comparison, ratio, root, integral and others","[{\"question\":\"What are the main derivative rules summarized in this sheet?\",\"answer\":\"It lists derivative shortcuts and core rules including product, quotient, and chain rule along with formulas for common functions such as sin, cos, tan, ln, and exponentials.\"},{\"question\":\"Which integral theorems and definite integral identities are included?\",\"answer\":\"The sheet includes the Fundamental Theorems of Calculus, along with start-plus-accumulation relations, Riemann sum guidance, and standard integral results for powers and basic trig/exponential/log forms.\"},{\"question\":\"What series expansions and convergence tests are provided?\",\"answer\":\"It provides elementary series plus Taylor and Maclaurin series structure, and includes multiple convergence/divergence tests such as ratio/limit comparison and other standard criteria.\"}]","BC Calc Memorization Sheet | PDF"]