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124exam3solutions

124exam3solutions - Math 124 Exam 3 Nov 3 2006 SOLUTIONS...

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Math 124 Exam 3 Nov. 3, 2006 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown. 2. Any numerical answers should be left in exact form, i.e, no decimal approximations. 3. You may use calculators. Good luck! 1) Find the derivative of g ( x ) = cosh 2 (9 x ) with respect to x . Solution: g ( x ) = 2 cosh(9 x ) sinh(9 x )(9) = 18 cosh(9 x ) sinh(9 x ) 2) Let h ( x ) = 1 + x . (a) Determine the local linearization of h ( x ) at a = 3. (b) Use the local linearization above to approximate h (3 . 1). (c) Is the approximation in part (b) an underestimate or an overestimate? Give a brief reason. Solutions: (a) First, note that h ( x ) = 1 2 (1 + x ) - 1 2 . Then h (3) = 1 4 . L ( x ) = h ( a ) + h ( a )( x - a ) = 2 + 1 4 ( x - 3) (b) h (3 . 1) L (3 . 1) = 2 + 1 4 (3 . 1 - 3) = 2 . 025 (c) The above approximation is an overestimate , as h is concave down at x = 3. Hence the tangent line lies above the function h . 1
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3) Consider the family of curves f ( x ) = ax + b x , where a and b are parameters. (a) Determine the critical points of this family in terms of the parameters. (b) Classify the critical points in part (a) as local minima, local maxima, or neither. Solutions: (a) f ( x ) = a - b x 2 = 0. Hence, a = b x 2 ax 2 = b x = ± b a (b) We compute a second derivative: f ( x ) = 2 b x 3 . Substituting the above critical points, we have
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