[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572877-105":53,"doc-detail-572877-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","ap-calculus-ab-2023-cheat-sheet","AP Calculus AB - 2023 Cheat Sheet","","AP Calculus AB – 2023 Cheat Sheet consolidates core AP Calculus AB concepts into a compact, problem-oriented reference. It covers key ideas across units including limits and continuity, differentiation rules (product/quotient/chain), estimation from graphs and tables, implicit differentiation, related rates, optimization, mean value theorems, and L’Hôpital’s Rule for indeterminate forms. It also summarizes integration as accumulation of change, Riemann sums, the Fundamental Theorem of Calculus, U-substitution guidance, and differential equations plus FRQ-style tips for applying these methods effectively.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/ap-calculus-ab-2023-cheat-sheet/572877/",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/ap-calculus-ab-2023-cheat-sheet/572877.png","ImageObject",442,249,{"name":88,"@type":89},"Levi","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is AROC and when should it be used in limits estimation?","Question",{"text":108,"@type":109},"AROC is the value f(x) approaches as x→c from both sides. When estimating f’(c) from a table, straddle both sides and use AROC; from a graph, use the slope of the tangent line.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which differentiation rules are summarized in the cheat sheet?",{"text":113,"@type":109},"It summarizes the product rule, quotient rule, and chain rule. It also includes guidance on estimating derivatives and notes that all differentiable functions are continuous, though not all continuous functions are differentiable.",{"name":115,"@type":106,"acceptedAnswer":116},"How does the cheat sheet describe using integration in AP Calculus AB?",{"text":117,"@type":109},"Integration is presented as accumulation of change, including how to interpret area under/above the x-axis. It also includes Riemann sums, the Fundamental Theorem of Calculus, and use of U-substitution for integrals.","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},572877,1792015512,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"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":9},7971461740909,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","♾AP Calculus AB – 2023 Cheat Sheet |  \n􀁿 See all AP Calc study guides |  @thinkfiveable  \n\n| 􀀿Unit 1\u003Cbr>Limits & Continuity |  | 􀁄Unit 2\u003Cbr>Fundamentals of Differentiation |  |  |  |  | 􀂫Unit 3 Composite, Implicit, & Inverse Functions |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| ●  is the value f (x) approaches when x → c from BOTH sides\u003Cbr>●  is the value that f (x) approaches when x → c from ONLY the right (if +) or left (if-) side |  | ●\u003Cbr>●●\u003Cbr>●\u003Cbr>●\u003Cbr>● | AROC = | |  |  | ●\u003Cbr>●\u003Cbr>●●\u003Cbr>● | Chain Rule: | |\n|  |  |  | When estimating f’(c) from a table, straddle both sides and use AROC, from a graph, slope of tangent line\u003Cbr>All differentiable functions are continuous, but not all continuous functions are differentiable |  |  |  |  |  |  |\n| ●●● | |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  | The chain rule is like unpeeling an onion, where you keep going from the outside in, you differentiate the outside function, plug in the inside function, and multiply by the derivative of the inside function.\u003Cbr>Implicit Differentiation: Differentiate each term with respect to the |  |\n|  |  |  | Power Rule:\u003Cbr>|  | |  |  |  |  |\n| ●\u003Cbr>● \u003Cbr>●Methods to algebraically simplify limits if you can’t directly plug in: Completing the square, Rationalization, Factoring\u003Cbr>●Order of growth rates from fastest to slowest: |  | ●\u003Cbr>●\u003Cbr>● | \u003Cbr>|  |  |  | ● | individual variables, and whenever you differentiate a y, multiply by \u003Cbr>Ex. \u003Cbr>Inverses: |  |\n| \u003Cbr>●For f (x)/g(x), if highest power of f > highest power of g: infinite limit DNE, if \u003C, HA at y=0, if =, HA at y = ratio of first terms\u003Cbr>●Continuity if \u003Cbr>●Removable: hole, Asymptote, and Jump: Piecewise where y-values different\u003Cbr>●IVT PROBLEMSր Write “Since f (x) is continuous on [a, b] and f (c) is between f (a) and f (b), by the IVT there is a cin (a, b) such that f (c) = f (c)” |  | ●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>● | \u003Cbr>Product Rule: |  |  | | ●\u003Cbr>●\u003Cbr>●●\u003Cbr>●\u003Cbr>● | \u003Cbr>Derivatives of inverse trig cofunctions are the negative of the derivative of the other 3 inverse trig functions\u003Cbr>Higher order derivatives: Just repeat!\u003Cbr>Second derivatives of implicit functions are functions of  |  |\n|  |  |  | Quotient Rule: |  |  | |  |  |  |\n\n♾AP Calculus AB – 2023 Cheat Sheet |  \n􀁿 See all AP Calc study guides |  @thinkfiveable  \n\n| 􀀮Unit 4\u003Cbr>Contextual Applications of Differentiation | ✨ Unit 5\u003Cbr>Analytical Applications of Differentiation | 􀀓Unit 6 Integration of Accumulation of Change |  |\n| --- | --- | --- | --- |\n| ● The derivative of a function is the rate of change of that function\u003Cbr>● If you are being asked about the rate of change of a rate of change, that’s basically the derivative of f’(x), orf”(x)\u003Cbr>● Particle motion: \u003Cbr>● Steps for Related Rates:\u003Cbr>1. Draw picture\u003Cbr>2. List knowns and unknowns\u003Cbr>3. Write an equation to model the situation ՄDO NOT PLUG IN STUFF THAT CHANGESՅ\u003Cbr>4. \u003Cbr>5. Substitute for changing values\u003Cbr>6. Solve for desired value\u003Cbr>● Linearization: \u003Cbr>● L’Hopital’s Rule ՄLHRՅր  ONLY IF  IS INDETERMINATE, that is  and\u003Cbr> are both either 0 or \u003Cbr>● Sometimes you may need to use LHR multiple times\u003Cbr>● Remember to plug in for the limit before doing LHR!! | ● MVT PROBLEMSր Write “Since f (x) is continuous on [a, b] and differentiable on\u003Cbr>(a. b), there exists a c in (a, b) where  by the MVT”\u003Cbr>● Rolle’s Theorem is MVT where f (b) = f (a)\u003Cbr>● EVT PROBLEMSր Write “Since f (x) is continuous on (a, b), by the EVT, there exists at least one local maximum and one local minimum on (a, b) .\u003Cbr>● Critical Points: Where f’(x) = 0 or undefined\u003Cbr>● Local Extrema: Point that is greater/less than surrounding points, always at critical points or endpoints\u003Cbr>● Global Extrema: Greatest or Least value of function\u003Cbr>● f’(x) > 0ր inc, f’(x) \u003C 0ր dec\u003Cbr>● If asking for whether the rate of change of f(x) is increasing or decreasing, this is asking for the sign of the second derivative!\u003Cbr>● Where f’(x) = 0, if f(x):  : lo","cbCaicAeHRdf2wzG","https://ap.wps.com/l/cbCaicAeHRdf2wzG","pdf",939318,"English","# Unit 1: Limits & Continuity\n# Unit 2: Fundamentals of Differentiation\n# Unit 3: Composite, Implicit, & Inverse Functions\n# Unit 4: Contextual Applications of Differentiation\n# Unit 5: Analytical Applications of Differentiation\n# Unit 6: Integration of Accumulation of Change\n# Unit 7: Differential Equations\n# Unit 8: Applications of Integration\n# FRQ Tips","[{\"question\":\"What is AROC and when should it be used in limits estimation?\",\"answer\":\"AROC is the value f(x) approaches as x→c from both sides. When estimating f’(c) from a table, straddle both sides and use AROC; from a graph, use the slope of the tangent line.\"},{\"question\":\"Which differentiation rules are summarized in the cheat sheet?\",\"answer\":\"It summarizes the product rule, quotient rule, and chain rule. It also includes guidance on estimating derivatives and notes that all differentiable functions are continuous, though not all continuous functions are differentiable.\"},{\"question\":\"How does the cheat sheet describe using integration in AP Calculus AB?\",\"answer\":\"Integration is presented as accumulation of change, including how to interpret area under/above the x-axis. It also includes Riemann sums, the Fundamental Theorem of Calculus, and use of U-substitution for integrals.\"}]","AP Calculus AB - 2023 Cheat Sheet | PDF",1791929856]