[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572342-105":53,"doc-detail-572342-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","ap-calculus-bc-2023-cheat-sheet","AP Calculus BC - 2023 Cheat Sheet","","AP Calculus BC 2023 cheat sheet condenses core exam concepts into quick-reference notes across Units 1–10. Coverage spans limits, continuity, differentiation rules (including chain, implicit, and inverse trig derivatives), and theorem-based reasoning such as IVT, MVT, and EVT. It also summarizes optimization, related rates, linearization, and integration topics including Riemann sums, FTC, and integral techniques like integration by parts and partial fractions. Additional sections include differential equations, applications of integration, parametric/polar/vector-valued ideas, and infinite sequences and series, plus FRQ tips and key formulas.",{"@graph":63,"@context":119},[64,80,102],{"@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-bc-2023-cheat-sheet/572342/",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-bc-2023-cheat-sheet/572342.png","ImageObject",442,249,{"name":88,"@type":89},"Aria Callaghan","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What does the sheet cover for differentiation rules and common theorem statements?","Question",{"text":109,"@type":110},"It summarizes differentiable vs continuous ideas and core differentiation rules, including chain rule, implicit differentiation, and derivatives of inverse trig functions. It also provides theorem writing guidance for IVT, MVT, and EVT.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are limits handled when direct substitution fails?",{"text":114,"@type":110},"It lists algebraic strategies such as completing the square, rationalization, and factoring to simplify limits that cannot be evaluated by plugging in directly.",{"name":116,"@type":107,"acceptedAnswer":117},"What integration and accumulation topics are included for exam practice?",{"text":118,"@type":110},"It includes the interpretation of integrals as area, Riemann sums (left/right/midpoint/trapezoidal), FTC, integration formulas with +C, and techniques such as integration by parts, tabular method, partial fraction decomposition, and improper integrals.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572342,1792022816,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":9},962084926284,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","♾AP Calculus BC – 2023 Cheat Sheet | 􀁿 See all AP Calc study guides | @thinkfiveable \n\n| 􀀿 Unit 1\u003Cbr>Limits & Continuity | | 􀁄 Unit 2 Fundamentals of Differentiation | | | | | | | 􀂫 Unit 3 Composite, Implicit, & Inverse Functions | | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| ●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●\u003Cbr>●●\u003Cbr>● | Order of growth rates from fastest to slowest:\u003Cbr>\u003Cbr>Methods to algebraically simplify limits if you can’t directly plug in: Completing the square, Rationalization, Factoring\u003Cbr>\u003Cbr>Continuity exists if \u003Cbr>Intermediate Value Theorem: 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 c in (a, b) such that\u003Cbr>f(c)=0” | ● All differentiable functions are continuous, but not all continuous functions are differentiable\u003Cbr>● Average Rate of Change = ● \u003Cbr>● Power Rule: \u003Cbr>● Sum/Difference Rule: \u003Cbr>● Product Rule: \u003Cbr>● Quotient Rule: \u003Cbr>● Other Differentiation Formulas: | | | | | | | ● Chain Rule: | | | | | | | |\n| | | | | | | | | | | | | | | | | |\n| | | | | | | | | | ● Implicit DiȨerentiation: DiȨerentiate each term with respect to the individual variables\u003Cbr>● Derivatives of an Inverse Trig Function:\u003Cbr>| | | | | | | |\n| | | | | | | | | | | | | | | | | |\n| | | | | | | | | | | | | | | | | |\n| | | | | | | | | | | | | | | | | |\n| 􀀮 Unit 4 Contextual Applications of Differentiation | | ✨ Unit 5\u003Cbr>Analytical Applications of Differentiation | | | | | | | 􀀓 Unit 6 Integration of Accumulation of Change | | | | | | | |\n| ● Particle Motion:\u003Cbr>○ Position =s (t)\u003Cbr>○ Velocity=v (t) =s’(t)\u003Cbr>○ Acceleration =a (t) =v’(t) =s’’(t)\u003Cbr>● If velocity is negative, the particle is moving to the left.\u003Cbr>● If velocity is positive, the particle is moving to the right.\u003Cbr>● If velocity and acceleration have the same sign, the particle is speeding up.\u003Cbr>● If velocity and acceleration have different signs, the particle is slowing down.\u003Cbr>● Steps for Related Rates:\u003Cbr>1. Draw a picture and label the picture (assign variables)\u003Cbr>2. List your knowns and unknown values\u003Cbr>3. Write an equation to model the situation.\u003Cbr>4. Take the derivative of both sides. Remember: d/dt\u003Cbr>5. Plug in known values and solve for desired values. DON’T FORGET UNITS!\u003Cbr>● Linearization: \u003Cbr>● If f(x)/g(x) are indeterminate that and are both equal to 0 or ∞ . Use L’Hopital’s Rule:\u003Cbr>| | ● Mean Value Theorem: Write “Since f(x) is continuous on [a, b] and diȨerentiable on (a. b), there exists a c in (a, b) such that\u003Cbr> by the MVT.\u003Cbr>● Extreme Value Theorem: 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 does not exist\u003Cbr>● f’(x) > 0: increasing; f’(x) \u003C 0: decreasing\u003Cbr>● First Derivative Test: where f’(x) = 0, if f(x): : local min; if f(x) : local max\u003Cbr>● Determining Concavity: f”(x) > 0: concave up; f”(x) \u003C 0: concave down; f”(x) = 0: inflection point\u003Cbr>● Second Derivative Test: f’(x) = 0, if f”(x) > 0: min; if f”(x) \u003C 0: max; if f”(x) = 0: indeterminate\u003Cbr>● Steps for Optimization:\u003Cbr>1. Draw picture\u003Cbr>2. Label your picture and assign variables\u003Cbr>3. Write an equation and use given information to find relationships among variables\u003Cbr>4. Find extrema (min/max) and evaluate the function | | | | | | | ● The integral is the area between the graph and the x-axis\u003Cbr>● Riemann Sum can be used to approximate area (includes left, right, midpoint, and trapezoidal sum) | | | | | | | |\n| | | | | | | | | | | | | | where | | and | |\n| | | | | | | | | | ● Fundamental Theorem of Calculus ՄFTCՅ -Definite Integrals\u003Cbr>● Integration Formulas ՄNEVER FORGET + CՅր\u003Cbr>| | | | | | | |\n| | | | | | | | | | | | | | | | | |\n| | | | | | | | | | | | | | | |","cbCait1rwicitiiL","https://ap.wps.com/l/cbCait1rwicitiiL","pdf",915795,"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# Unit 9 Parametric Equations, Polar Coordinates, & Vector Valued Functions\n# Unit 10 Infinite Sequences & Series\n# FRQ Tips\n# Formulas","[{\"question\":\"What does the sheet cover for differentiation rules and common theorem statements?\",\"answer\":\"It summarizes differentiable vs continuous ideas and core differentiation rules, including chain rule, implicit differentiation, and derivatives of inverse trig functions. It also provides theorem writing guidance for IVT, MVT, and EVT.\"},{\"question\":\"How are limits handled when direct substitution fails?\",\"answer\":\"It lists algebraic strategies such as completing the square, rationalization, and factoring to simplify limits that cannot be evaluated by plugging in directly.\"},{\"question\":\"What integration and accumulation topics are included for exam practice?\",\"answer\":\"It includes the interpretation of integrals as area, Riemann sums (left/right/midpoint/trapezoidal), FTC, integration formulas with +C, and techniques such as integration by parts, tabular method, partial fraction decomposition, and improper integrals.\"}]","AP Calculus BC - 2023 Cheat Sheet | PDF",1791926852]