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Calculus Cheat Sheet
Elementary Calculus II (Math 2007)
Carleton University
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Limits & Derivatives Cheat Sheet
Properties of Limits lim 𝑥→𝑎 𝑐𝑓𝑥 = 𝑐 lim 𝑥→𝑎 𝑓(𝑥) lim 𝑥→𝑎 [𝑓𝑥 ±𝑔𝑥 ] = lim 𝑥→𝑎 𝑓(𝑥)± lim 𝑥→𝑎 𝑔(𝑥) lim 𝑥→𝑎 [𝑓𝑥 𝑔𝑥 ] = lim 𝑥→𝑎 𝑓(𝑥) lim 𝑥→𝑎 𝑔(𝑥) lim 𝑥→𝑎 𝑓𝑥 𝑔𝑥 = lim 𝑥→𝑎 𝑓(𝑥) lim 𝑥→𝑎 𝑔(𝑥) 𝑖𝑓 lim 𝑥→𝑎 𝑔(𝑥) ≠ 0 lim 𝑥→𝑎 [𝑓𝑥 ]𝑛= lim 𝑥→𝑎 𝑓𝑥 𝑛 Limit Evaluations At ±∞ lim 𝑥→+∞ 𝑒𝑥= ∞and lim 𝑥→−∞ 𝑒𝑥= 0 lim 𝑥→∞ ln𝑥 = ∞and lim 𝑥→0+ ln𝑥 = −∞ ifr > 0: lim 𝑥→∞ 𝑐 𝑥𝑟 = 0 ifr > 0& ∀𝑥 > 0 𝑥𝑟∈ ℝ ∶ lim 𝑥→−∞ 𝑐 𝑥𝑟 = 0 lim 𝑥→±∞ 𝑥𝑟= ∞forevenr lim 𝑥→+∞ 𝑥𝑟= ∞and lim 𝑥→−∞ 𝑥𝑟= −∞foroddr L’Hopital’sRule If lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = 0 0 or ±∞ ±∞ then lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = lim 𝑥→𝑎 𝑓′(𝑥) 𝑔′(𝑥) Common Derivatives 𝑑 𝑑𝑥 𝑥 = 1 𝑑 𝑑𝑥 [𝑎𝑓𝑥 ] = 𝑎 𝑑 𝑑𝑥 [𝑓𝑥 ] 𝑑 𝑑𝑥 𝑎𝑥 = 𝑎 𝑑 𝑑𝑥 𝑎𝑥𝑛 = 𝑛𝑎𝑥𝑛− 𝑑 𝑑𝑥 𝑐 = 0 𝑑 𝑑𝑥 [𝑓𝑥 ]𝑛= 𝑛[𝑓𝑥 ]𝑛−1𝑓′(𝑥) 𝑑 𝑑𝑥 1 𝑥𝑛 = −𝑛𝑥−𝑛+1 = − 𝑛 𝑥𝑛+ Derivative Definition 𝑑 𝑑𝑥 𝑓𝑥 = 𝑓′𝑥 = lim ℎ→ 𝑓𝑥 +ℎ −𝑓(𝑥) ℎ Product Rule 𝑓𝑥 𝑔𝑥 ′= 𝑓′𝑥 𝑔𝑥 +𝑓 𝑥 𝑔′(𝑥) Quotient Rule 𝑑 𝑑𝑥 𝑓𝑥 𝑔𝑥 = 𝑓′𝑥 𝑔𝑥 −𝑓𝑥 𝑔′(𝑥) [𝑔𝑥 ] 2 Chain Rule 𝑑 𝑑𝑥 𝑓𝑔𝑥 = 𝑓′𝑔𝑥 𝑔′(𝑥) Derivatives of Trigonometric Functions 𝑑 𝑑𝑥 sin𝑥 = cos𝑥 𝑑 𝑑𝑥 sec𝑥 = sec𝑥tan𝑥 𝑑 𝑑𝑥 cos𝑥 = −sin𝑥 𝑑 𝑑𝑥 csc𝑥 = −csc𝑥cot𝑥 𝑑 𝑑𝑥 tan𝑥 = sec 2 𝑥 𝑑 𝑑𝑥 cot𝑥 = −csc 2 𝑥 Derivatives of Exponential & Logarithmic Functions 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑥 𝑑 𝑑𝑥 𝑎𝑥 = 𝑎𝑥ln𝑎 𝑑 𝑑𝑥 ln|𝑥| = 1 𝑥 𝑑 𝑑𝑥 ln𝑥 = 1 𝑥 ,𝑥 > 0 𝑑 𝑑𝑥 log𝑎𝑥 = 1 𝑥ln𝑎 𝑑 𝑑𝑥 ln𝑓(𝑥) = 𝑓′(𝑥) 𝑓(𝑥) 𝑑 𝑑𝑥 𝑒𝑓(𝑥) = 𝑓′(𝑥)𝑒𝑓(𝑥) 𝑑 𝑑𝑥 𝑎𝑓𝑥 = 𝑎𝑓(𝑥)ln𝑎𝑓′(𝑥) 𝑑 𝑑𝑥 𝑓𝑥𝑔𝑥 = 𝑓𝑥𝑔𝑥 𝑔𝑥 𝑓′𝑥 𝑓𝑥 +ln𝑓𝑥 𝑔′𝑥 @SmartGirlStudy | SmartGirlStudy Derivatives of Inverse Trig Functions 𝑑 𝑑𝑥 sin−1𝑥 = 1 1−𝑥 2 𝑑 𝑑𝑥 sec−1𝑥 = 1 |𝑥| 𝑥 2 − 1 𝑑 𝑑𝑥 cos−1𝑥 = − 1 1− 𝑥 2 𝑑 𝑑𝑥 csc−1𝑥 = − 1 |𝑥| 𝑥 2 − 1 𝑑 𝑑𝑥 tan−1𝑥 = 1 1+ 𝑥 2 𝑑 𝑑𝑥 cot−1𝑥 = − 1 1+𝑥 2 Basic Properties of Derivatives 𝑐𝑓𝑥 ′= 𝑐[𝑓′𝑥 ] 𝑓𝑥 ±𝑔𝑥 ′= 𝑓′(𝑥)± 𝑔′(𝑥) Derivatives of Hyperbolic Functions 𝑑 𝑑𝑥 sinh𝑥 =cosh𝑥 𝑑 𝑑𝑥 sech𝑥 = −coth𝑥csch𝑥 𝑑 𝑑𝑥 cosh𝑥 = sinh𝑥 𝑑 𝑑𝑥 csch𝑥 = −tanh𝑥sech𝑥 𝑑 𝑑𝑥 tanh𝑥 = 1 −tanh 2 𝑥 𝑑 𝑑𝑥 coth𝑥 = −1−coth 2 𝑥
Integration Cheat Sheet
Integration Properties න 𝑎 𝑏 𝑐𝑓𝑥 𝑑𝑥 = 𝑐න 𝑎 𝑏 𝑓𝑥 𝑑𝑥 න 𝑎 𝑏 𝑓𝑥 ± 𝑔𝑥 𝑑𝑥 = න 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 ± න 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥 න 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥 = 0 න 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 = −න 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥 න 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 + න 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥 = න 𝑎 𝑐 𝑓𝑥 𝑑𝑥 Integration by Substitution න 𝑎 𝑏 𝑓𝑔𝑥 𝑔′(𝑥) 𝑑𝑥 = න 𝑔(𝑎) 𝑔(𝑏) 𝑓𝑢 𝑑𝑢 where𝑢 = 𝑔𝑥 and𝑑𝑢 = 𝑔′𝑥 𝑑𝑥 Common Integrals න𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶 න𝑥𝑛𝑑𝑥 = 𝑥𝑛+ 𝑛 + 1
- 𝐶 න 1 𝑥 𝑑𝑥 = ln|𝑥|+ 𝐶 නln𝑥𝑑𝑥 = 𝑥 ln𝑥 − 𝑥 + 𝐶 න𝑒𝑥𝑑𝑥 = 𝑒𝑥+ 𝐶 න𝑏𝑥𝑑𝑥 = 𝑏𝑥 ln𝑏
- 𝐶 න 1 𝑎𝑥 + 𝑏 𝑑𝑥 = 1 𝑎 ln|𝑎𝑥 + 𝑏| + 𝐶 න 1 𝑥 2 − 𝑎 2 𝑑𝑥 = 1 2𝑎 ln 𝑥 − 𝑎 𝑥 + 𝑎 න 1 𝑥 2 ± 𝑎 2 𝑑𝑥 = ln𝑥 + 𝑥 2 ± 𝑎 2 Integration by Parts න𝑢𝑑𝑣 = 𝑢𝑣 −න𝑣 𝑑𝑢, where𝑣 =න𝑑𝑣 orන𝑓𝑥 𝑔′(𝑥) 𝑑𝑥 = 𝑓𝑥 𝑔𝑥 − න𝑓′𝑥 𝑔(𝑥)𝑑𝑥 Fundamental Theorem of Calculus න 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 = [𝐹𝑥 ]𝑎𝑏= 𝐹𝑏 −𝐹𝑎 where𝑓iscontinuouson 𝑎,𝑏 and𝑓′= 𝐹 Integrals of Trigonometric Functions නcos𝑥 𝑑𝑥 = sin𝑥 + 𝐶 නcsc𝑥cot𝑥𝑑𝑥 = −csc𝑥 + 𝐶 නsin𝑥 𝑑𝑥 = −cos𝑥 + 𝐶 නsec𝑥tan𝑥𝑑𝑥 = sec𝑥 + 𝐶 නsec 2 𝑥𝑑𝑥 = tan𝑥 + 𝐶 නcsc 2 𝑥𝑑𝑥 = −cot𝑥 + 𝐶 නcot𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶 නcsc𝑥 𝑑𝑥 = ln|csc𝑥 − cot𝑥|+ 𝐶 නsinh𝑥 𝑑𝑥 = cosh𝑥 + 𝐶 නcosh𝑥 𝑑𝑥 = sinh𝑥 + 𝐶 නtan𝑥𝑑𝑥 = ln|sec𝑥|+ 𝐶 නsec𝑥𝑑𝑥 = ln|sec𝑥 + tan𝑥| + 𝐶 න 1 𝑎 2 + 𝑥 2 𝑑𝑥 = 1 𝑎 tan− 𝑥 𝑎
- 𝐶 න 1 𝑎 2 − 𝑥 2 𝑑𝑥 = sin− 𝑥 𝑎
- 𝐶 @SmartGirlStudy SmartGirlStudy Definite Integral Definition න 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 = lim 𝑛→∞ 𝑘= 𝑛 𝑓𝑥𝑘 ∆𝑥 where∆𝑥 = 𝑏 −𝑎 𝑛 𝑎𝑛𝑑 𝑥𝑘= 𝑎 +𝑘∆𝑥 Integrals of Symmetric Functions If𝑓 iseven𝑓−𝑥 = 𝑓𝑥 ,then න −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥 = 2න 0 𝑎 𝑓𝑥 𝑑𝑥 If𝑓isodd 𝑓−𝑥 = −𝑓𝑥 ,then න −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥 = 0 Partial Fractions 𝑅(𝑥)𝑄(𝑥)𝐴 1 𝑎 1 𝑥 + 𝑏 1
- 𝐴 2 𝑎 2 𝑥 + 𝑏 2
- ⋯+ 𝐴𝑛 (𝑎𝑛𝑥 + 𝑏𝑛) , where 𝑄𝑥 = 𝑎 1 𝑥 + 𝑏 1 𝑎 2 𝑥 + 𝑏 2 ...(𝑎𝑛𝑥 + 𝑏𝑛) 𝑅(𝑥)𝑄(𝑥)𝐴 1 𝑎 1 𝑥 + 𝑏 1
- 𝐴 2 (𝑎 1 𝑥 + 𝑏 1 ) 2
- ⋯+ 𝐴𝑛 𝑎 1 𝑥 + 𝑏 1 𝑛 where alinear factorof 𝑄𝑥 isrepeated 𝑛 times 𝑅(𝑥)𝑄(𝑥)𝐴𝑥 + 𝐵 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑄𝑥 hasafactor𝑎𝑥 2 + 𝑏𝑥 + 𝑐,where 𝑏 2 − 4𝑎𝑐 < 0 𝑅(𝑥)𝑄𝑥𝐴 1 𝑥 + 𝐵 1 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
- 𝐴 2 𝑥 + 𝐵 2 (𝑎𝑥 2 + 𝑏𝑥 + 𝑐) 2
- ⋯+ 𝐴𝑛𝑥 + 𝐵𝑛 𝑎𝑥 2 + 𝑏𝑥 + 𝑐𝑛 where 𝑄𝑥 hasafactor 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ,where 𝑏 2 − 4𝑎𝑐 < 0
Calculus Cheat Sheet
Course: Elementary Calculus II (Math 2007)
University: Carleton University
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