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2004 Fall - 125FinalFall04short math125

2004 Fall - 125FinalFall04short math125 - MATH 125 FINAL...

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MATH 125 FINAL EXAM (common exam) December 2004 1. (5 points each) Compute the following limits, if they exist. You may not use l’Hˆ opital’s rule, if you know what that is. You must also justify your answer. (a) lim x + 3 x 2 + 1 (1 - x )(2 x - 1) (b) lim x 0 x cot 7 x (c) lim x 1 5 - x - 4 x x 2 - 1 2. (6 points each) Find dy dx of the following functions. (You do not need to simplify your answer.) (a) y = sin ( ln(1 + x 100 ) ) (b) y = x e e x (c) y = x 0 (1 + e - t 2 ) dt (d) y = tan x 1 + sec x 3. (6 points each) Evaluate the following integrals. (You do not need to simplify your answer.) (a) 5 x 2 + 1 x + 2 cos 2 x - sin x dx (b) 1 0 x cos(1 + x 2 ) dx (c) 5 x 1 + 5 x dx 4. (12 points) Knowing that 4 81 = 3, use a linear approximation to estimate 4 85. 5. (15 points) Find an equation of the tangent line to the curve xe 2 x - y 2 ln y = 0 at the point (1 , e ). 6. (7 points each) Consider the function f ( x ) = x cos 1 x if x = 0 0 if x = 0 (a) Is f continuous at x = 0? Explain your answer. 1
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(b) Is f differentiable at x = 0? Explain your answer.
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