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Unformatted text preview: Exam 2 Solutions October 29, 2010 Total: 60 points Problem 1: Find the derivative of each of the following functions g ( x ) = log(cos( x 2 )) x sin x h ( x ) = e Solution For the first problem, we invoke the chain rule twice. Thus, d 1 dx (log(cos( x 2 ))) = cos( x 2 ) ( sin( x 2 )) 2 x = 2 x tan( x 2 ) . The second problem requires the chain rule and the product rule. We imme- diately get, d ( e x sin x ) = e x sin x 1 sin x + x cos x . dx 2 x Problem 2: Consider the function log x g ( x ) = . 2 x Determine the behavior of g in a neighborhood of x = 1. Specifically, is the function increasing or decreasing? Is it convex or concave? Justify your answers. d 1 d 2 Solution We determined dx log x = x for x > and dx x = 2 x = for x > 0. Thus, we can use the quotient rule to determine g ( x ) ,g ( x ) away from x = 0. A quick calculation yields 1 2 log x g ( x ) = 3 x and g ( x ) = 5 + 6 log x ....
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