final_fa00 - Math 191 FINAL EXAM Fall 2000 SHOW ALL WORK...

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Math 191 FINAL EXAM Fall 2000 SHOW ALL WORK. CIRCLE YOUR ANSWERS. CLOSED BOOK. NO CALCULATORS. 1. (25 pts) Let f ( x ) = x x = e x ln x , x > 0 (a) Compute lim x 0 + f ( x ) if it exists. (b) Locate and identify the critical points of f ( x ) in x > 0. (c) Find the absolute maximum and absolute minimum of f ( x ) in x > 0 if they exist. (d) Find the linearization of f at x = 1. (e) Solve the equation x x = 1 . 01 approximately using one iteration of Newton’s method (assuming that the initial guess is x 0 = 1). 2. (10 pts) Solve the initial value problem dy dx = ( y 2 - 4) sin x , y (0) = 3 You may leave your answer as an implicit equation for the function y ( x ). 3. (10 pts) (a) Use the Trapezoid Rule with four subintervals to approximate the integral Z 1 - 1 dt 1 + t 2 . (b) Is the approximation you obtained in (a) an over-estimate or an under-estimate of the true value of the integral? Explain with the aid of a sketch. 4. (15 pts) You want to place a ladder with one end on the ground and the other end against the vertical wall of a tall building. Between you and the wall, there is a fence that is 4
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