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Final Formula Sheet
Université de Montréal
Uploaded: A year ago
Contributor: blithering
Category: Calculus
Type: Equation Sheet
Rating: B (1)
Filename:   Final Formula Sheet.pdf (28.75 kB)
Page Count: 2
Credit Cost: 1
Views: 227
Last Download: N/A
Transcript
Formula Sheet for Final Exam Differentiation Rules ′ ⎛ f ⎞ g⋅ f′− f ⋅g′ Product: (f ⋅g)′= f ⋅g′+g⋅ f′ Quotient: ⎜⎜ ⎟⎟ = 2 ⎝ g ⎠ g Exp, Log Derivatives d x x d 1 a = a lna (loga x) = dx dx xlna Trig Derivatives d d d 2 sinx =cosx cosx −sinx tanx =sec x dx dx dx d 2 d d cotx = −csc x secx =secxtanx cscx =−cscxcotx dx dx dx d −1 1 d −1 1 −1 1 (sin x)= (cos x)= − (csc )=− dx 2 2 2 1−x dx 1−x x −1 d −1 1 d −1 1 d −1 1 (sec x)= (tan x)= (cot x)=− dx 2 2 2 x x −1 dx 1+ x dx 1+ x Hyperbolic Functions x −x x −x e −e 1 e +e sinhx= cschx= coshx= 2 sinhx 2 1 sinhx coshx sechx= tanhx= cothx= coshx coshx sinhx Derivatives of Hyperbolic Functions d d (sinhx)=coshx (cschx)=−cschx⋅cothx dx dx d d (coshx)=sinhx (sechx)=−sechx⋅tanhx dx dx d 2 d 2 (tanhx)=sech x (cothx)=−csch x dx dx Derivatives of Inverse Hyperbolic Functions d −1 1 d −1 1 d −1 1 (sinh x)= (cosh x)= (tanh x)= dx 2 2 2 1+x dx x −1 dx 1−x Table of Integrals n+1 n x x x x dx= +C , (n ≠ −1) e dx = e +C ∫ n+1 ∫ x x a 1 a dx = +C, a ≠1 dx = ln| x |+C ∫ lna ∫ x sin(x)dx=−cosx+C cos(x)dx =sin x+C ∫ ∫ 2 2 sec (x)dx=tanx+C csc (x)dx=−cotx+C ∫ ∫ sec(x)tan(x)dx=secx+C csc(x)cot(x)dx=−cscx+C ∫ ∫ Summation Formulas n n n(n +1) c = nc ∑ i ∑ = i=1 i=1 2 n n 2 2 n(n +1)(2n +1) 3 ⎡n(n +1)⎤ i ∑ = i = ∑ ⎢ ⎥ i=1 6 i=1 ⎣ 2 ⎦ Miscellaneous Formulas st f (x+h)− f (x) 1 Principles Derivatives: f ′(x) = lim h→0 h Linear Approximation: f (x) ≈ f (a) + f ′(a)(x − a) f (b)− f (a) Mean Value Theorem: f ′(c) = b − a b 1 Average Value: fave = f (x)dx b − a ∫ a b Mean Value Theorem for Integrals: f (x)dx = f (c)(b−a) ∫ a • NOTE: If needed, the formulas for the inverse hyperbolic functions (e.g. −1 2 sinh x=ln(x+ x +1)) will be provided within the question. In addition, any formulas for surface area and volume of geometric shapes (which sometimes come up in related rates/optimization questions) will be provided.
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Comments (1)
A year ago
Thanks for Calc 1 formula sheet!


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