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2005 Fall (short) - 125FinalFall05shrt math125

# 2005 Fall (short) - 125FinalFall05shrt math125 - MATH 125...

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MATH 125, FINAL EXAM (common) December 2005 1. (6 points each) Calculate the following limits. a) lim x 4 x - 4 x - 2 . b) lim x 1 e x - e x - 1 . c) lim x 0 3 x + x 2 sin x . 2. (7 points each) Find dy dx . a) y = ln(2 x 2 - 3 x ). b) y = 2 x - 3 e x +1 . c) y = x sin x (Use logarithmic differentiation). d) y = tan x 0 t 2 + 4 dt for - π 2 x π 2 . 3. (8 points each) Evaluate the following integrals: (a) 2 1 x 2 ( x - 2) 2 5 dx . (b) sin x x dx . (c) Find the area of the region bounded by the curves y = 0, y = xe x 2 , x = 0 and x = 1. 4. Consider the following function and its first and second derivative: f ( x ) = x x 2 + 4 f ( x ) = - x 2 - 4 ( x 2 + 4) 2 f ( x ) = 2 x ( x 2 - 12) ( x 2 + 4) 3 . a) (5 points) Find the critical numbers of f . b) (10 points) Determine where f is increasing, where f is decreasing, and find the local maxima and minima of f . c) (5 points) Find the asymptotes of f . f ( x ) = x x 2 + 4 f ( x ) = - x 2 - 4 ( x 2 + 4) 2 f ( x ) = 2 x ( x 2 - 12) ( x 2 + 4) 3 . d) (5 points) Find the inflection points of f and determine where f is concave upwards and downwards. e) (8 points) Draw a rough sketch of the graph of f . 1

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2 5. (12 points) a) Find the absolute maximum and minimum of f ( x ) = xe - 3 x on the interval [ - 1 , 1]. b) (8 points) For what values of c is the function f ( x ) continuous
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2005 Fall (short) - 125FinalFall05shrt math125 - MATH 125...

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