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124exam2solutions

# 124exam2solutions - Math 124 Exam 2 Oct 10 2006 SOLUTIONS...

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Math 124 Exam 2 Oct. 10, 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) [21 points] Find the derivatives of each of the following functions with respect to the variable indicated. There is no need to simplify your answers. (a) f ( x ) = 2 x 3 - 1 2 x with respect to x. (b) g ( t ) = 2 t 3 - t 2 + w t 2 with respect to t. (c) h ( r ) = π r + r π + π e with respect to r. (d) k ( x ) = (2 x 3 + e x ) 3 2 x 3 + 1 with respect to x. (e) w ( θ ) = θ sin( θ 2 ) with respect to θ. (f) f ( z ) = ( z + arctan z ) e with respect to z. (g) g ( x ) = (ln x ) π - ln( x π ) with respect to x. Solutions: (a) f ( x ) = 2 x 3 - 1 2 x - 1 , so f ( x ) = 6 x 2 + 1 2 x - 2 (b) Rewriting g , we have g ( t ) = 2 t - 1 + wt - 2 , so g ( t ) = 2 - 2 wt - 3 (c) h ( r ) = π r ln π + πr π - 1 (d) Using the quotient rule, we have k ( x ) = 3(2 x 3 + e x ) 2 (6 x 2 + e x ) 2 x 3 + 1 - (2 x 3 + e x ) 3 1 2 (2 x 3 + 1) - 1 2 (6 x 2 ) 2 x 3 + 1 (e) Using the product rule, we have w ( θ ) = sin( θ 2 ) + θ cos( θ 2 )(2 θ ) (f) f ( z ) = e ( z + arctan z ) e - 1 1 + 1 1 + z 2 (g) g ( x ) = π (ln x ) π - 1 1 x - πx π - 1 x π 1

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2) [14 points] Consider the function y = f ( x ) graphed below. Notice that f ( x ) is defined for - 5 < x < 6, except at x = 2. What x -values in the domain of f have the following properties? Estimate as best as you can, if necessary.
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124exam2solutions - Math 124 Exam 2 Oct 10 2006 SOLUTIONS...

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