quiz3solutions

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1 MATH 104 QUIZ III SOLUTIONS CLAY SHONKWILER = ln(ln(ln y dy/dx . x (1) Let )). Find Using repeated applications of the chain rule: Answer: d 1 dy · = [ln(ln x )] ln(ln dx dx ) x [ ] 1 1 d = · (ln x ) · ln(ln ln x ) x dx [ ] 1 1 1 · · = ) ln(ln x ln x x 1 = . x ln(ln ln x x ) (2) Evaluate the integral ∫ 1 /x e dx. 2 x 4 − 1 /x . Then du = . Hence, we can re-write Let u = 1 dx Answer: 2 x the integral as ( ) ∫ ∫ 1 − 1 − 1 − 1 − 1 − 1 u /x /x u 1 = du = e + e e e + C = C. dx 2 4 x 4 4 4 2 x = x y . Find dy/dx . (3) Let 2 x 2 2 ln( x x ) x ln x Recall that x = y = e Answer: . Therefore, = e ] [ d dy 2 x 2 ln x e x · ln x = dx dx ] [ 1 2 x 2 = x x + 2 x ln · x x 2 x x = [ x + 2 x ln x ] 2 x +1 ] = [1 + 2 ln x x . DRL 3E3A, University of Pennsylvania E-mail address : [email protected] 1

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