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It also provides key antiderivative and definite integral definitions, the fundamental theorem of calculus (parts I and II), and common integration properties and examples, focusing on standard rule-based computation and boundary conditions.","| \u003Cbr>CALCULUS LIMITS AND DERIVATIVES |  |  |  |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| LIMIT PROPERTIES |  |  |  |  |  |  |  | DERIVATIVE FORMULAS |  | DERIVATIVE NOTATION |\n| Assume that the limits of 􀝂(􀝔) and 􀝃(􀝔) exist as 􀝔 approaches 􀜽 . lim [􀝂(􀝔) ± 􀝃 (􀝔)] = lim 􀝂(􀝔) ± lim 􀝃 (􀝔)\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔\u003Cbr>lim [􀜿􀝂 (􀝔)] = 􀜿 lim 􀝂(􀝔)\u003Cbr>􀯫→􀯔 􀯫→􀯔\u003Cbr>l􀯫 [􀝂 (􀝔)􀝃 (􀝔)] = (l􀯫 􀝂 (􀝔)) (l􀯫 􀝃 (􀝔)) l􀯫 [􀝂􀝃((􀝔􀝔))] = l~~􀯫~~l􀯫~~ ~~􀝂􀝃((􀝔􀝔)) if l􀯫i 􀝃 (􀝔) ≠ 0 l􀯫 [􀝂 (􀝔)]􀯡 = [l􀯫 􀝂 (􀝔)]􀯡\u003Cbr>l􀯫 􀳙√􀝂(􀝔) = 􀳙√l􀯫 􀝂 (􀝔) |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀜿) = 0\u003Cbr>􀝀􀝀􀝔 (􀝔) = 1\u003Cbr>􀝀􀝀􀝔 (sin 􀝔) = cos 􀝔􀝀􀝀􀝔 (cos 􀝔) = − sin 􀝔􀝀􀝀􀝔 (tan 􀝔) = sec2 􀝔􀝀􀝀􀝔 (sec 􀝔) = sec 􀝔 tan 􀝔􀝀􀝀􀝔 (cot 􀝔) = − csc2 􀝔\u003Cbr>􀝀􀝀􀝔 (csc 􀝔) = − csc 􀝔 cot 􀝔\u003Cbr> 􀝀  (sin−1 􀝔) =  1 􀝀􀝔 √1 − 􀝔 2\u003Cbr> 􀝀  (cos−1 􀝔) = −  1 􀝀􀝔 √1 − 􀝔 2  􀝀   1 􀝀􀝔 (tan−1 􀝔) = 1 + 􀝔 2 􀝀􀝀􀝔 (cot−1 􀝔) = − ~~ ~~1~~ ~~􀝔~~ ~~2\u003Cbr> 􀝀  (sec−1 􀝔) =  1 􀝀􀝔 |􀝔|√􀝔2 − 1\u003Cbr> 􀝀  (csc−1 􀝔) = −  1 􀝀􀝔 |􀝔|√􀝔2 − 1  􀝀 \u003Cbr>(􀝁 􀯫 ) = 􀝁 􀯫􀝀􀝔\u003Cbr>􀝀􀝀􀝔 (􀜾􀯫 ) = 􀜾 􀯫 ln 􀜾􀝀􀝀􀝔 (ln 􀝔) = 1􀝔 (􀝔 > 0)\u003Cbr>􀝀􀝀􀝔 (log􀯕 􀝔) = 􀝔~~ ~~l1n~~ ~~􀜾 (􀝔 > 0) |  | If 􀝕 = 􀝂 (􀝔), then the following are equivalent notations for the derivative.􀝀􀝀􀝕􀝔 = 􀝕 ′ = 􀝂 ′(􀝔) = 􀝀􀝀􀝂􀝔 = 􀝀􀝀􀝔 (􀝂(􀝔)) |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | DERIVATIVE DEFINITION |\n|  |  |  |  |  |  |  |  |  |  |  􀝀 􀝂 (􀝔) = 􀝂 ′(􀝔) = lim 􀝂 (􀝔 + ℎ) − 􀝂(􀝔)􀝀􀝔 ℎ→0 ℎ |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | POWER RULE |\n|  |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀝔􀯡 ) = 􀝊􀝔􀯡−1 |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n| FUNDAMENTAL LIMITS |  |  |  |  |  |  |  |  |  | PRODUCT RULE |\n| lim 􀜿 = 􀜿􀯫→􀯔 |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀝂(􀝔)􀝃 (􀝔)) = 􀝂 (􀝔)􀝃 ′(􀝔) + 􀝃 (􀝔)􀝂 ′(􀝔) |\n|  |  |  |  |  |  |  |  |  |  |  |\n| lim 1 􀯫→∞ 􀝔 |  | = 0 and lim 1 = 0 􀯫→−∞ 􀝔 |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | \u003Cbr>QUOTIENT RULE |\n| l􀯫 􀝔1􀯣 = 0 for any 􀝌 > 0 lim 􀝔 􀯣 = ∞ for any 􀝌 > 0 􀯫→∞\u003Cbr>lim 􀝔 􀯣 = ∞ for even 􀝌 and lim 􀝔 􀯣 = −∞ for odd 􀝌\u003Cbr>􀯫→−∞ 􀯫→−∞\u003Cbr>lim 􀝁 􀯫 = ∞ and lim 􀝁 􀯫 = 0\u003Cbr>􀯫→∞ 􀯫→ − ∞\u003Cbr>lim ln 􀝔 = ∞ and lim ln 􀝔 = −∞􀯫→∞ 􀯫→0+\u003Cbr>lim tan−1 􀝔 = 􀟨 and lim tan−1 􀝔 = − 􀟨􀯫→∞ 2 􀯫→−∞ 2 |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀝂􀝃~~ ~~((􀝔􀝔))) = 􀝃~~ ~~(􀝔)􀝂~~ ~~′(􀝔[􀝃)()􀝂]2(􀝔)􀝃~~ ~~′(􀝔) |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | CHAIN RULE |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀝂 (􀝃 (􀝔))) = 􀝂 ′(􀝃 (􀝔))􀝃 ′(􀝔) |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | DERIVATIVE PROPERTIES |\n|  |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 (􀝂(􀝔) + 􀝃 (􀝔)) = 􀝀􀝀􀝔 􀝂 (􀝔) + 􀝀􀝀􀝔 􀝃 (􀝔)\u003Cbr>􀝀􀝀􀝔 (􀝂(􀝔) − 􀝃 (􀝔)) = 􀝀􀝀􀝔 􀝂 (􀝔) − 􀝀􀝀􀝔 􀝃 (􀝔)\u003Cbr>􀝀􀝀􀝔 (􀜿􀝂 (􀝔)) = 􀜿 ⋅ 􀝀􀝀􀝔 􀝂 (􀝔) |\n|  |  |  |  |  |  |  |  |  |  |  |\n| lim 􀰏→0 | sin 􀟠􀟠 |  | = 1 and lim 􀰏→0 |  | 􀟠\u003Cbr>sin 􀟠 | = 1 |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n| lim 􀰏→0 | 1 − cos 􀟠\u003Cbr>􀟠 |  |  | = 0 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | |\n| L’HOPITAL’S RULE |  |  |  |  |  |  |  |  |  | CHAIN RULE FORMS |\n| 􀝂 (􀝔) 0 ±∞ 􀝂 (􀝔) 􀝂′(􀝔)\u003Cbr>If lim ~~ ~~ = ~~ ~~ or ~~ ~~ then lim ~~ ~~ = lim~~ ~~′~~ ~~􀯫→􀯔 􀝃 (􀝔) 0 ±∞ 􀯫→􀯔 􀝃 (􀝔) 􀯫→􀯔 􀝃 (􀝔)\u003Cbr>If lim 􀝂(􀝔)􀝃 (􀝔) = 0 ⋅ (±∞) then lim 􀝂(􀝔)􀝃 (􀝔) = lim  􀝂 (􀝔)  = lim  􀝃 (􀝔) \u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 1/􀝃 (􀝔) 􀯫→􀯔 1/􀝂(􀝔)\u003Cbr>If lim [􀝂 (􀝔)]􀯚 (􀯫) = 00 or ∞0 or 1∞ then lim [􀝂 (􀝔)]􀯚 (􀯫) = lim 􀝁 ln[􀯙(􀯫)]􀳒 (􀳣) = 􀝁 l􀳣 􀯚 (􀯫) ln[􀯙(􀯫)]􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |  |  |  |  |  |  |  |  |  | 􀝀􀝀􀝔 [􀝃 (􀝔)]􀯡 = 􀝊 [􀝃 (􀝔)]􀯡−1􀝃 ′(􀝔)\u003Cbr>􀝀􀝀􀝔 (􀝁 􀯚 (􀯫)) = 􀝁 􀯚 (􀯫)􀝃 ′(􀝔)\u003Cbr>􀝀􀝀􀝔 (􀜾 􀯚 (􀯫)) = 􀜾 􀯚 (􀯫)􀝃 ′(􀝔) ln 􀜾􀝀􀝀􀝔 ln (􀝃 (􀝔)) = 􀝃~~ ~~1(􀝔) 􀝃 ′(􀝔) = 􀝃􀝃′((􀝔􀝔))􀝀􀝀􀝔 log􀯕 (􀝃 (􀝔)) = 􀝃~~ ~~􀝃(􀝔′)(􀝔ln)~~ ~~􀜾􀝀􀝀􀝔 sin (􀝃 (􀝔)) = 􀝃 ′(􀝔) cos (􀝃 (􀝔))\u003Cbr>􀝀􀝀􀝔 cos (􀝃 (􀝔)) = −􀝃′(􀝔) sin(􀝃 (􀝔))\u003Cbr>􀝀􀝀􀝔","cbCaimzKfmy8gNYj","https://ap.wps.com/l/cbCaimzKfmy8gNYj","pdf",297160,2,"English","en",105,"# Calculus Limits and Derivatives\n## Limit Properties\n## Derivative Definition\n## Power Rule\n## Fundamental Limits\n## Product Rule\n## Quotient Rule\n## Chain Rule\n## Derivative Properties\n## L’Hôpital’s Rule\n## The Squeeze Theorem\n# Calculus Integrals\n## Definite Integral Definition\n## Fundamental Theorem of Calculus, Part I\n## Fundamental Theorem of Calculus, Part II\n## Basic Integration Properties","[{\"question\":\"What limit properties and derivative formulas are summarized in the sheet?\",\"answer\":\"It lists standard limit properties (e.g., linearity and power/constant behaviors) and common derivative formulas, along with derivative notation conventions for equivalent forms.\"},{\"question\":\"Which differentiation rules are included?\",\"answer\":\"The sheet includes the derivative definition plus the power rule, product rule, quotient rule, and chain rule (including chain rule forms).\"},{\"question\":\"What integration topics are covered?\",\"answer\":\"It covers common indefinite integral formulas, the definite integral definition, the fundamental theorem of calculus (parts I and II), basic integration properties, and related examples such as logarithmic and trigonometric integrals.\"}]","Calculus Formula Reference Sheet - Limits and Derivatives - Integrals | 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limit properties and derivative formulas are summarized in the sheet?","Question",{"text":123,"@type":124},"It lists standard limit properties (e.g., linearity and power/constant behaviors) and common derivative formulas, along with derivative notation conventions for equivalent forms.","Answer",{"name":126,"@type":121,"acceptedAnswer":127},"Which differentiation rules are included?",{"text":128,"@type":124},"The sheet includes the derivative definition plus the power rule, product rule, quotient rule, and chain rule (including chain rule forms).",{"name":130,"@type":121,"acceptedAnswer":131},"What integration topics are covered?",{"text":132,"@type":124},"It covers common indefinite integral formulas, the definite integral definition, the fundamental theorem of calculus (parts I and II), basic integration properties, and related examples such as logarithmic and trigonometric integrals.","https://schema.org",{"og:url":94,"og:type":135,"og:title":59,"og:site_name":106,"og:description":60},"article",{"robots":137,"canonical":94},"index,follow",{"doc_id":55,"site_id":69},1792017536]