CalcReviewSolutions

CalcReviewSolutions - Geometric Series Do the series...

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Solutions STA 505 Review of Calculus Worksheet Derivatives For each of the following expressions, make sure you can take both the first and second derivatives with respect to the variable x . 1. 2 1 5 5 2 x x x x x π + + = + - (Power Rule) 2. ( 29 17 17 17 x x e e = (Chain Rule) 3. ( 29 ( 29 2 3 3 2 3 3 5 10 15 5 2 3 x x x x x e xe x e xe x = + = + (Product, Power and Chain Rules) 4. ( 29 ( 29 2 2 1 2 ln 2 x x x x = = (Chain Rule) Easier solution: use rules for logs to simplify first… ( 29 2 ln 2ln x x = Then, ( 29 2 2ln x x = 5. ( 29 ( 29 1 ln 1 x x x e e e = = Easier solution: Note that ( 29 ln x e x = . So its derivative is 1. 6. ( 29 4 5 5 15 14 3 14 2 3 14 x x x x x - - = - (Chain rule) Integrals Compute the following integrals. 1. 2 2 5 5 2 x x e dx e c = + 2. 2 2 3 3 4 2 1 1 3 4 2 5 2 5 3 4 3 3 5 2 2 5 2 2 2 ... 4 3 3 4 3 3 x x dx x x x + + = + + = + + - + + = 3. ( 29 1 x x x x x x xe dx xe e dx xe e c e x c = - = - + = - + (integration by parts) 4. ( 29 ( 29 ( 29 ( 29 0 0 0 0 lim lim lim 0 0 1 1 t t x x x x t t t t t xe dx xe dx xe e te e e - - - - →∞ →∞ - - →∞ = = + = + - - = - - =
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5. ( 29 2 1 2 1 3 2 4 2 2 0 1 0 1 1 2 0 1 2 3 0 5 7 5 7 4 2 5 7 20 14 4 2 75 21 75 7 8 6 8 2 x y x y x x xy x y dydx xy x y dx x x dx x x = = = = = = + = + = - + - = + = -
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Unformatted text preview: Geometric Series Do the series converge or diverge? If they converge, what do they converge to? 1. 1 1 1 1 1 1 1 1 4 4 4 1 4 i i i i = = =- =- = - 2. 1 5 20 5 1 4 3 1 4 i i = = = - 3. 1 5 4 i i = diverges because r > 1. 4. What is the Taylor series expansion for x e ? ! n x n x e n = = for all x . Rules for Exponents and Logs Simplify if possible using exponent and logarithmic identities. 1. ( 29 ( 29 ( 29 ln ln ln ab a b = + 2. a b a b e e e + = 3. ( 29 ln a b + cant not be reduced any further. 4. ( 29 3 ln 3ln x x = 5. ( 29 ln x e x =...
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CalcReviewSolutions - Geometric Series Do the series...

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