[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-314487-105":59,"doc-detail-314487-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","ap-calculus-abbc-formula-and-concept-cheat-sheet","AP Calculus AB/BC Formula and Concept Cheat Sheet","","AP Calculus AB/BC formula and concept reference sheet covering limits for continuous functions, rational functions, and infinities, plus special trigonometric limits and L’Hospital’s Rule for indeterminate forms. It includes continuity and types of discontinuities, the Intermediate Value Theorem, average rate of change, and the definition and alternate definition of the derivative. The sheet further summarizes differentiability rules, derivative rules by category, explicit/implicit differentiation, tangent and normal lines, and mean value theorem variants such as Rolle’s theorem. It also covers particle motion and related rates.",{"@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/ap-calculus-abbc-formula-and-concept-cheat-sheet/314487/",{"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/ap-calculus-abbc-formula-and-concept-cheat-sheet/314487.png","ImageObject",300,407,{"name":92,"@type":93},"Aditya","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you evaluate limits of rational functions at a vertical asymptote?","Question",{"text":113,"@type":114},"Determine whether the limit exists, diverges to ±∞, or yields a removable discontinuity. Use the conditions based on common factors between numerator and denominator and whether x approaches the value that makes the denominator zero.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What are the key steps for applying L’Hospital’s Rule?",{"text":118,"@type":114},"First identify an indeterminate form after rewriting the limit. Then apply the rule by differentiating numerator and denominator and re-evaluating the limit with the resulting quotient.",{"name":120,"@type":111,"acceptedAnswer":121},"When is a function differentiable at x = c, and how does that relate to continuity?",{"text":122,"@type":114},"If a function is differentiable at x = c, it must be continuous there. A function can be continuous but not differentiable at c when the graph has a cusp or sharp point, a vertical tangent, or an endpoint.","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},314487,1789900933,{"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":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},962085564549,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","AP Calculus AB/BC Formula and Concept Cheat Sheet  \nLimit of a Continuous Function  \nIff(x) is a continuous function for all real numbers, then lim 􀝂(􀝔) = 􀝂 (􀜿)  \n􀯫→􀯖  \nLimits of Rational Functions  \nA. Iff(x) is a rational function given by 􀝂(􀝔) = 􀯣􀯤((􀯫􀯫)) ,such that 􀝌(􀝔) and 􀝍 (􀝔) have no common factors, and c is a real number such that 􀝍(􀜿) = 0, then  \nI. lim 􀝂(􀝔) does not exist 􀯫→􀯖  \nII. lim 􀝂(􀝔) = ±∞  x = c is a vertical asymptote 􀯫→􀯖  \nB. Iff(x) is a rational function given by 􀝂(􀝔) = 􀯣􀯤((􀯫􀯫)), such that reducing a common factor between 􀝌 (􀝔) and 􀝍 (􀝔) results in the agreeable function k(x), then  \nl􀯫i 􀝂 (􀝔) = l􀯫i 􀯣􀯤((􀯫􀯫)) = l􀯫i 􀝇 (􀝔) = 􀝇 (􀜿) Hole at the point (􀜿, 􀝇 (􀜿))  \nLimits of a Function as x Approaches Infinity  \nIff(x) is a rational function given by (􀝔) = 􀯣􀯤((􀯫􀯫)) , such that 􀝌(􀝔) and 􀝍 (􀝔) are both polynomial functions, then  \nA. If the degree of p(x) > q(x), lim 􀝂(􀝔) = ∞  \n􀯫→∞  \nB. If the degree of p(x) \u003C q(x), lim 􀝂(􀝔) = 0  y = 0 is a horizontal asymptote  \n􀯫→∞  \nC. If the degree of p(x) = q(x), lim 􀝂(􀝔) = 􀜿, where c is the ratio of the leading coefficients.  \n􀯫→∞   \ny = c is a horizontal asymptote  \nSpecial Trig Limits  \nA. lim sin 􀯔􀯫 = 1 B. lim 􀯔􀯫  = 1 C. lim 1−cos 􀯔􀯫 = 0  \n􀯫→0 􀯔􀯫 􀯫→0 sin 􀯔􀯫 􀯫→0 􀯔􀯫  \n\n| L’Hospital’s Rule\u003Cbr>If results l􀯫i 􀝂 (􀝔) or l􀯫􀝂 (􀝔) results in an indeterminate form ( 00 ,  , ∞ − ∞ , 0 ∙ ∞ , 00 , 1∞ , ∞0 ) , and 􀝂 (􀝔) = 􀯣􀯤((􀯫􀯫)) , then |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| lim 􀝂(􀝔) = lim\u003Cbr>􀯫→􀯖 􀯫→􀯖 | 􀯣 (􀯫)\u003Cbr>􀯤 (􀯫) | = lim\u003Cbr>􀯫→􀯖 | 􀯣′(􀯫)\u003Cbr>􀯤′(􀯫) | and lim 􀝂(􀝔) = lim\u003Cbr>􀯫→∞ 􀯫→∞ | 􀯣 (􀯫)\u003Cbr>􀯤 (􀯫) | = lim\u003Cbr>􀯫→∞ | 􀯣′(􀯫)\u003Cbr>􀯤′(􀯫) |\n\nThe Definition of Continuity  \nA function 􀝂(􀝔) is continuous at c if  \nI. lim 􀝂(􀝔) exists 􀯫→􀯖  \nII. 􀝂 (􀜿) exists  \nIII. lim 􀝂(􀝔) = 􀝂 (􀜿)􀯫→􀯖  \n\n| Types of Discontinuities\u003Cbr>Removable Discontinuities (Holes)\u003Cbr>| I. lim 􀝂(􀝔) = 􀜮 (the limit exists)􀯫→􀯖\u003Cbr>II. 􀝂 (􀜿) is undefined |\n| --- | --- |\n| Non-Removable Discontinuities (Jumps and Asymptotes)\u003Cbr>A. Jumps\u003Cbr>lim 􀝂(􀝔) = 􀜦􀜰􀜧 because lim 􀝂 (􀝔) ≠ lim 􀝂 (􀝔)\u003Cbr>􀯫→􀯖 􀯫→􀯖 − 􀯫→􀯖+\u003Cbr>\u003Cbr>B. Asymptotes (Infinite Discontinuities) |  |\n| | lim 􀝂(􀝔) = ±∞􀯫→􀯖 |\n\nIntermediate Value Theorem  \nIf f is a continuous function on the closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one value of c on [a, b] such that f(c) = k. In other words, on a continuous function, iff(a)\u003C f(b), any y – value greater than f(a) and less than f(b) is guaranteed to exists on the function f.  \n| Average Rate of Change\u003Cbr>The average rate of change, m, of a function fon the interval [a, b] is given by the slope of the secant line. |  |\n| --- | --- |\n| | 􀝉 = 􀯙 (􀯕)−􀯙(􀯔)􀯕−􀯔 |\n\nDefinition of the Derivative  \nThe derivative of the function f, or instantaneous rate of change, is given by converting the slope of the secant line to the slope of the tangent line by making the change is x, Δx or h, approach zero.  \n􀝂 ′(􀝔) = lim 􀯙 (􀯫+ℎ)−􀯙(􀯫)ℎ→0 ℎ  \nAlternate Definition  \n􀝂 ′(􀜿) = lim 􀯙 (􀯫)−􀯙(􀯖)􀯫→􀯖 􀯫−􀯖  \nDifferentiability and Continuity Properties  \nA. Iff(x) is differentiable at x = c, then f(x) is continuous at x = c.  \nB. Iff(x) is not continuous at x = c, then f(x) is not differentiable at x = c.  \nC. The graph off is continuous, but not differentiable at x = c if:  \nI. The graph has a cusp or sharp point at x = c  \nII. The graph has a vertical tangent line at x = c  \nIII. The graph has an endpoint at x = c  \nBasic Derivative Rules  \nGiven c is a constant,  \n| Derivatives of Trig Functions |\n| --- |\n| |\n\n\n| Derivatives of Inverse Trig Functions |\n| --- |\n| |\n|  |\n\n\n| Derivatives of Exponential and Logarithmic Functions |\n| --- |\n| |\n|  |\n\nExplicit and Implicit Differentiation  \nA. Explicit Functions: Function y is written only in terms of the variable x (􀝕 = 􀝂 (􀝔)). Apply derivatives rules normally.  \nB. Implicit Differentiation: An expression representing the graph of a curve in terms of both variables x and y.  \nI. Differentiate both sides of the equation with respect to x. (terms","cbCailoALwW1SHnA","https://ap.wps.com/l/cbCailoALwW1SHnA","pdf",1485951,25,"English","# Limit of a Continuous Function\n# Limits of Rational Functions\n# Limits as x Approaches Infinity\n# Special Trig Limits\n# L’Hospital’s Rule\n## The Definition of Continuity\n## Types of Discontinuities\n# Intermediate Value Theorem\n# Average Rate of Change\n# Definition of the Derivative\n## Differentiability and Continuity Properties\n# Basic Derivative Rules\n# Explicit and Implicit Differentiation\n# Tangent Lines and Normal Lines\n# Mean Value Theorem for Derivatives\n## Rolle’s Theorem\n# Particle Motion\n# Related Rates","[{\"question\":\"How do you evaluate limits of rational functions at a vertical asymptote?\",\"answer\":\"Determine whether the limit exists, diverges to ±∞, or yields a removable discontinuity. Use the conditions based on common factors between numerator and denominator and whether x approaches the value that makes the denominator zero.\"},{\"question\":\"What are the key steps for applying L’Hospital’s Rule?\",\"answer\":\"First identify an indeterminate form after rewriting the limit. Then apply the rule by differentiating numerator and denominator and re-evaluating the limit with the resulting quotient.\"},{\"question\":\"When is a function differentiable at x = c, and how does that relate to continuity?\",\"answer\":\"If a function is differentiable at x = c, it must be continuous there. A function can be continuous but not differentiable at c when the graph has a cusp or sharp point, a vertical tangent, or an endpoint.\"}]","AP Calculus AB/BC Formula and Concept Cheat Sheet | PDF",63]