PracticePrelim2Solutions

PracticePrelim2Solutions - Math 1110 Practice Prelim 2...

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Math 1110 Practice Prelim 2 Solutions Please not that what follows are solutions intended for checking your work. Despite being correct, some of these solutions would not receive full credit were they to be given on an exam. In particular, the solutions to the true/false questions would not receive full credit on the exam because although correct counterexamples are given, they are not fully explained. 1. For each of the following functions, determine the derivative with respect to x . (a) f ( x ) = x 2 + 7 x 3 + 3 (b) f ( x ) = 3 x (c) f ( x ) = (4 x + 10) 7 (d) f ( x ) = 2 x 3 x 2 +1 Solution: (a) f 0 ( x ) = 2 x + 21 x 2 (b) f 0 ( x ) = 3 x ln 3 (c) f 0 ( x ) = 28(4 x + 10) 6 (d) f 0 ( x ) = 2(3 x 2 + 1) - 12 x 2 (3 x 2 + 1) 2 2. For each of the following functions, determine the derivative with respect to x . (a) f ( x ) = ln(ln x ) (b) f ( x ) = x ( x 2 ) Solution: (a) f 0 ( x ) = 1 x ln x (b) f ( x ) = e x 2 ln x f 0 ( x ) = e x 2 ln x (2 x ln x + x ) = x ( x 2 ) (2 x ln x + x ) 3. Two non-negative numbers add up to 20. Find those numbers if the product of one and the square root of the other is to be as large as possible. Solution: Maximize x y subject to x + y = 20 and x, y 0. In other words, maximize f ( x ) = x 20 - x for x [0 , 20].
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