[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573236-105":53,"doc-detail-573236-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","calculus-bc-formulas-section-1-limits-section-2-derivatives-section-3-integrals","Calculus BC Formulas - Section 1 Limits, Section 2 Derivatives, Section 3 Integrals","","Calculus BC formulas organized by topic, covering key limit principles (including the squeeze theorem), continuity and the intermediate value theorem, and derivative fundamentals. The derivative section summarizes differentiation rules for polynomials, exponentials, logarithms, and trigonometric functions, plus L’Hospital’s rule, the mean value theorem, extreme value theorem, and the second-derivative test. The integrals and differential equations section includes integral concepts, integration rules, and applications such as the disk method for volume of solids of revolution.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/calculus-bc-formulas-section-1-limits-section-2-derivatives-section-3-integrals/573236/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/calculus-bc-formulas-section-1-limits-section-2-derivatives-section-3-integrals/573236.png","ImageObject",442,249,{"name":88,"@type":89},"Mia  ","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the squeeze theorem conclude about a limit?","Question",{"text":108,"@type":109},"It states that if g(x) ≤ f(x) ≤ h(x) for x ≠ c and both lim g(x) and lim h(x) as x→c equal L, then lim f(x) as x→c is also L.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which conditions define continuity of a function at x = c?",{"text":113,"@type":109},"A function f is continuous at x = c when f(c) is defined, the limit lim(x→c) exists, and that limit equals f(c).",{"name":115,"@type":106,"acceptedAnswer":116},"How is the disk method used to find volume of a solid of revolution?",{"text":117,"@type":109},"Rotate the region about a horizontal line y = c. Use disks where the radius is r, the thickness is dx, and r is the distance from the axis of revolution to the function f(x), then integrate across the interval.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573236,1792067516,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":25,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":47},687207024478,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Calculus BC Formulas  \nSection 1: Limits  \nsqueeze theorem  \n\n| Conditions | Conclusion |\n| --- | --- |\n| g(x) ≤ ƒ(x) ≤ h(x) for x ≠ c | lim ƒ(x) = L x → c |\n| lim g(x) = L and lim h(x) = L\u003Cbr>x → c x → c |  |\n\n| Intermediate Value Theorem (IVT)\u003Cbr>If a function ƒ is continuous on the interval [a , b] and k is a number between ƒ(a) and ƒ(b), then there is at least one x-value c between a and b such that ƒ(c) = k. |  |\n| --- | --- |\n| ƒ(b)\u003Cbr>ƒ(a) \u003C k \u003C ƒ(b)\u003Cbr> k\u003Cbr>\u003Cbr>ƒ(a) \u003Cbr>c3 c1 c2 a a \u003C c \u003C b b\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>y | x |\n| \u003Cbr>Any continuous function connecting (a , 􀀀(a)) and (b , 􀀀(b)) must pass through every y-value between 􀀀(a) and 􀀀(b) at least once. |  |\n\nA function ƒ(x) is continuous at x = c if all of the following conditions are met:  \n• ƒ(c) is defined  \n• lim x → c  \n• lim  \nƒ(x) exists  \nƒ(x) = ƒ(c)  \nx → c  \nThe graph of a continuous function has no \"gaps.\"  \ny  \ny  \nSection 2: Derivatives  \nDifferentiation rules  \n\n| Constant | d~~d~~x [c] = 0 |\n| --- | --- |\n| Power | d~~d~~x [xn] = nxn − 1 |\n| Natural exponential | d~~d~~x [ex ] = ex |\n| Exponential | d~~d~~x [ax ] = (ln a)ax |\n| Natural log | ~~dd~~x [ln(x )] = ~~ ~~1x |\n| Constant multiple | d~~d~~x [cƒ(x )] = cƒ ' (x) |\n| Sum and difference | d~~d~~x [ƒ(x) ± g(x)] = ƒ'(x) ± g'(x) |\n\nDerivatives of trigonometric functions  \nd~~d~~x [sinx] = cosx  \n\n| d~~d~~x [cosx] = −sinx |\n| --- |\n| d~~d~~x [tanx] = sec2x |\n| d~~d~~x [cotx] = −csc2x |\n| d~~d~~x [cscx] = −csc(x) cot(x) |\n\nd~~d~~x [secx] = sec(x) tan(x)  \n|  position   velocity   acceleration  | x(t) v(t) = x ′(t) a(t) = x ″(t) | differentiate |\n| --- | --- | --- |\n\nUse L'Hospital's Rule to find the limit of the ratio of two differentiable functions ƒg(~~(~~xx)~~)~~ as x approaches c. If direct substitution produces one of the indeterminate forms ~~00~~ or ~~ ~~~~ ~~ , then differentiate the numerator ƒ and denominator g independently.  \nƒ(x) ƒ '(x) lim ~~ ~~ = lim ~~  ~~x→c g(x) x→c g'(x)  \nL'Hospital's Rule also applies to limits as x→∞ or x→−∞ .  \nMean Value Theorem  \n\n| Conditions | Conclusion |\n| --- | --- |\n| ƒ is continuous on [a, b] | For some c in (a, b):\u003Cbr>ƒ(b) − ƒ(a)ƒ ′(c) = b − a\u003Cbr>instantaneous rate average rate of change at x = c of change on [a, b]\u003Cbr>|\n| ƒ is differentiable on (a, b) |  |\n\n| Extreme Value Theorem (EVT)\u003Cbr>If a function ƒ is continuous on the closed interval [a,b], then ƒ is guaranteed to attain an absolute minimum and absolute maximum value on [a,b] .\u003Cbr>y |  |\n| --- | --- |\n| O\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>maximum\u003Cbr>a b | x |\n\nsecond derivative test for critical point at x = c  \n\n| If ƒ″(c) \u003C 0 | If ƒ″(c) = 0 | If ƒ″(c) > 0 |\n| --- | --- | --- |\n| ƒ (c) is a | test is | ƒ(c) is a |\n| \u003Cbr>relative maximum | \u003Cbr>inconclusive | \u003Cbr>relative minimum |\n\nSection 3: Integrals and Differential Equations  \n\n| \u003Cbr>The base length bn of each rectangle is the distance between the endpoints of the subinterval, and the height hn is the function value at the left endpoint.\u003Cbr>|  |\n| --- | --- |\n| \u003Cbr>\u003Cbr>\u003Cbr>ƒ(x1)\u003Cbr>h\u003Cbr>h 2\u003Cbr>1\u003Cbr>O a = x1 b1 x2 b2 x3 \u003Cbr>|   x |\n\n\n| \u003Cbr>|  |\n| --- | --- |\n| y\u003Cbr>\u003Cbr>ƒ\u003Cbr>ƒ(m2)\u003Cbr>|   x |\n\nFundamental Theorem of Calculus  \nlimit of right Riemann sum  \n| integrate\u003Cbr>integrate | a (t) acceleration v (t) velocity s (t) position |\n| --- | --- |\n\nSecond FTC  \nFundamental Theorem of Calculus  \n(alternate form)  \nIntegration rules  \nConstant  \nPower  \nConstant multiple  \nSum and difference  \nNatural exponential  \nNatural log  \n| The following are properties of definite integrals, where functions fand g are continuous on the closed interval [a, b] and a, b, and k are constants.\u003Cbr>|  | \u003Cbr>improper integral\u003Cbr>|\n| --- | --- | --- |\n|  |  |  |\n|  |  | \u003Cbr>integration by parts\u003Cbr>udv = uv − v du |\n|  |  |  |\n|  |  |  |\n| Integrals of trigonometric functions\u003Cbr>|  | \u003Cbr>logistic differential equation\u003Cbr>|\n|  |  |  |\n|  |  | \u003Cbr>logistic differential equation\u003Cbr>~~dd~~P~~t~~ = kP(a − P) |\n|  |  |  |\n|  |  | \u003Cbr>Euler's method equation\u003Cbr>yn + 1 = yn + ƒ′(xn ) (Δx)\u003Cbr>\u003Cb","cbCaibmuM9H05CUJ","https://ap.wps.com/l/cbCaibmuM9H05CUJ","pdf",1072938,"English","# Section 1: Limits\n## squeeze theorem\n# Section 2: Derivatives\n## Differentiation rules\n## Mean Value Theorem\n## Extreme Value Theorem (EVT)\n## second derivative test for critical point\n# Section 3: Integrals and Differential Equations","[{\"question\":\"What does the squeeze theorem conclude about a limit?\",\"answer\":\"It states that if g(x) ≤ f(x) ≤ h(x) for x ≠ c and both lim g(x) and lim h(x) as x→c equal L, then lim f(x) as x→c is also L.\"},{\"question\":\"Which conditions define continuity of a function at x = c?\",\"answer\":\"A function f is continuous at x = c when f(c) is defined, the limit lim(x→c) exists, and that limit equals f(c).\"},{\"question\":\"How is the disk method used to find volume of a solid of revolution?\",\"answer\":\"Rotate the region about a horizontal line y = c. Use disks where the radius is r, the thickness is dx, and r is the distance from the axis of revolution to the function f(x), then integrate across the interval.\"}]","Calculus BC Formulas - Section 1 Limits, Section 2 Derivatives, Section 3 Integrals | PDF",1791935439]