[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-462565-105":59,"doc-detail-462565-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","the-ap-calculus-ab-cheat-sheet","The AP Calculus AB Cheat Sheet","","Condensed study reference for AP Calculus AB covering key concepts across Units 1–8, with emphasis on limit behavior, continuity, and solving derivative-related problems. Includes core strategies for evaluating limits (substitution, factoring, rationalizing, squeeze theorem, L'Hôpital’s), continuity and discontinuity criteria, asymptotes, and interval-based proof prompts for FRQ. Summarizes derivative rules, chain/implicit differentiation, related rates workflow, inverse function differentiation, and differential equation concepts like slope fields and motion analysis.",{"@graph":69,"@context":122},[70,84,105],{"@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/the-ap-calculus-ab-cheat-sheet/462565/",{"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/the-ap-calculus-ab-cheat-sheet/462565.png","ImageObject",300,407,{"name":92,"@type":93},"Aria","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you evaluate limits when direct substitution leads to 0/0 or ∞/∞?","Question",{"text":112,"@type":113},"Start with direct substitution when it is valid; if you get 0/0 or ∞/∞, use factoring, rationalizing (conjugates), or L'Hôpital’s rule by differentiating the numerator and denominator and taking the new limit.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What conditions must hold for a function to be continuous at x=a?",{"text":117,"@type":113},"Continuity at x=a requires: (1) the limit lim(x→a) f(x) exists, (2) f(a) is defined, and (3) lim(x→a) f(x) = f(a). All three are needed.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the workflow for related rates problems?",{"text":121,"@type":113},"Identify the variables and write a constraint equation, differentiate both sides with respect to t, substitute known values, and solve for the unknown rate, keeping units consistent.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},462565,1790764501,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":81,"preview_page_count":4},2336464648322,"https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488","[The AP Calculus AB Cheat Sheet](The AP Calculus AB Cheat Sheet www.albert.io)[ www.albert.io](The AP Calculus AB Cheat Sheet www.albert.io)  \n Unit 1: Limits & Continuity (10–12%)  Unit 2: Differentiation Basics (10–12%)  Unit 3: Chain, Implicit, Inverse (9–13%)  Unit 4: Contextual Applications (10–15%)   \n| Limit Strategies & Computation |  |\n| --- | --- |\n| Direct substitution first: plug in, works if no 0/0 form. If 0/0, try factor, rationalize, conjugate, or L'Hôpital's.\u003Cbr>L'Hôpital's rule: if lim f/g = 0/0 or ∞/∞, then lim f/g = lim f'/g' (differentiate top & bottom) .\u003Cbr>Case Approach\u003Cbr>∞/∞ rational Divide by highest power |  |\n| 0/0 rational | Factor and cancel |\n| 0/0 radicals | Rationalize (conjugate) |\n| Squeeze Theorem | f ≤ g ≤ h, lim f = lim h |\n| Continuity & Discontinuity |  |\n| Continuous at x = a iff (1) lim(x →a) f(x) exists, (2) f(a) defined,(3) lim = f(a) . Need all 3.\u003Cbr>Types: removable (hole), jump (both sides exist but ≠), infinite (vertical asymptote) .\u003Cbr>Intermediate Value Theorem: f continuous [a,b] & f(a) ≠ f(b) → f hits every value between.\u003Cbr>✓ IVT proves solutions exist. FRQ: 'show equation has solution on interval.'\u003Cbr>Asymptotes & Limits at Infinity |  |\n| Horizontal: lim(x →±∞) f(x) = L → y = L. Rationals: compare degrees. Numer higher → none.\u003Cbr>Vertical: denom = 0 & numer ≠ 0. Graph: f → ±∞ at x = a. Oblique: use long division if numer degree = denom + 1.\u003Cbr>⚠ Say 'diverges to ∞', not 'limit = ∞ ' (∞ is not a number) .\u003Cbr>Piecewise limits: evaluate left & right separately. If different, limit DNE.\u003Cbr>FRQ prep: prove continuity (show 3 conditions) or prove limit DNE (left ≠ right) .\u003Cbr>Unit 5: Analytical Applications (15–18%) |  |\n| Critical Points & Extrema |  |\n\n\n| Definition & Meaning |  |  | Chain Rule\u003Cbr>Composite: y = f(g(x)) . Then dy/dx = f'(g(x)) · g'(x)(outside' × inside') .\u003Cbr>Example: d/dx[sin(x²)] = cos(x²) · 2x. Stacked: multiply all three derivatives.\u003Cbr>Pattern Derivative\u003Cbr>eᵍ⁽ˣ⁾ eᵍ⁽ˣ⁾ · g'(x)\u003Cbr>ln(g(x)) g'(x)/g(x) [g(x)]ⁿ n[g(x)]ⁿ⁻¹ · g'(x) Implicit Differentiation Method: differentiate both sides w. r.t. x. Treat y as\u003Cbr>function: d/dx[y] = dy/dx.\u003Cbr>Second Implicit Deriv: To find d2~~ ~~ydx~~2~~ , differentiate and *substitute* the original dydx expression back\u003Cbr>dy dx into\u003Cbr>your new equation.\u003Cbr>Related rates: implicit differentiate, then solve for rate (dx/dt, dy/dt) . Boat, ladder, shadow problems.✓ Related rates: draw, label vars, write constraint, diff w. r.t. t, plug knowns, solve.\u003Cbr>Inverse Functions If y = f⁻¹ (x), then f(y) = x. Diff: 1 = f'(y) · dy/dx → dy/dx = 1/f'(y) .\u003Cbr>Standard: d/dx[arcsin x] = 1/√(1-x²), d/dx[arctan x] = 1/(1+x²), d/dx[ln x] = 1/x.\u003Cbr>Stacked chain: d/dt[sin(x²)] where x=x(t) → cos(x²)·2x·dx/dt. Multiply three terms.\u003Cbr>Implicit on y-terms: d/dx[y²] = 2y·dy/dx (chain rule) . Apply on every y.\u003Cbr>Related rates workflow: identify vars, write constraint, diff w. r.t. t, solve for unknown rate. Unit 7: Differential Equations (6–12%)\u003Cbr>Slope Fields & Methods | Motion Analysis\u003Cbr>Chain: x(t)=pos → v=dx/dt → a=d²x/dt². Interpret: v>0 moving right, a>0 speeding up.\u003Cbr>Condition Meaning\u003Cbr>v > 0, a > 0 Speeding up forward v > 0, a \u003C 0 Slowing down forward v \u003C 0, a \u003C 0 Speeding up backward Distance vs displacement: split at v=0, sum |Δx| . Speed=|v| always pos. Displacement=x(b)-x(a) . Speeding up/slowing down: Speed increases when v(t) and a(t) have the same sign. Speed decreases when signs are opposite.\u003Cbr>Related Rates Setup: identify vars, write constraint (V=πr²h), differentiate w. r.t. t, plug in knowns, solve for rate. Units matter: if dr/dt in cm/s, dV/dt in cm³/s. State answer with units.\u003Cbr>Multi-variable chains: if V=πr²h & both r, h change w/t, diff each term via chain rule.\u003Cbr>Marginal Analysis\u003Cbr>Marginal cost = dC/dx. Marginal revenue = dR/dx. Profit = R-C.\u003Cbr>Max profit: set dP/dx = 0. Check 2nd derivative: d² P/dx² \u003C 0 confirms max. Check endpoints. |  |\n| --- | --- | --- | --- | --- | --- |\n| Derivative: f'(a) = lim(h →0)","cbCaihv6urqpmlNM","https://ap.wps.com/l/cbCaihv6urqpmlNM","pdf",102411,"English","# Unit 1: Limits & Continuity\n## Limit strategies & computation\n## Continuity & discontinuity\n## Asymptotes & limits at infinity\n# Unit 2-4: Differentiation & applications\n## Derivative definition & basic rules\n## Chain rule and composite functions\n## Implicit differentiation and higher-order implicit derivatives\n## Related rates and inverse functions\n# Unit 5-8: Analytical and integration applications\n## Critical points & extrema\n## Optimization and marginal analysis\n## Differential equations and motion analysis\n## Integration & accumulation\n## Applications & FRQ tips","[{\"question\":\"How do you evaluate limits when direct substitution leads to 0/0 or ∞/∞?\",\"answer\":\"Start with direct substitution when it is valid; if you get 0/0 or ∞/∞, use factoring, rationalizing (conjugates), or L'Hôpital’s rule by differentiating the numerator and denominator and taking the new limit.\"},{\"question\":\"What conditions must hold for a function to be continuous at x=a?\",\"answer\":\"Continuity at x=a requires: (1) the limit lim(x→a) f(x) exists, (2) f(a) is defined, and (3) lim(x→a) f(x) = f(a). All three are needed.\"},{\"question\":\"What is the workflow for related rates problems?\",\"answer\":\"Identify the variables and write a constraint equation, differentiate both sides with respect to t, substitute known values, and solve for the unknown rate, keeping units consistent.\"}]","The AP Calculus AB Cheat Sheet | PDF"]