191
FINAL Math EXAM
Fall 2000
SHOW ALL WORK. CIRCLE YOUR ANSWERS. CLOSED BOOK. NO CALCULATORS. 1. (25 pts) Let f (x) = xx = ex ln x , x > 0 (a) Compute lim f (x) if it exists. +
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(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 xx = 1.01 approximately using one iteration of Newton's method (assuming that the initial guess is x0 = 1). 2. (10 pts) Solve the initial value problem dy = (y 2 - 4) sin x , dx y(0) = 3
You may leave your answer as an implicit equation for the function y(x). 3. (10 pts) dt . -1 1 + t2 (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. (a) Use the Trapezoid Rule with four subintervals to approximate the integral 4. (15 pts) You want to place a ladder with one end on the ground and the other end against the 1 vertical wall of a tall building. Between you and the wall, there is a fence that is 4 2 ft high and 10 2 ft from the building. 3 (a) Make a sketch of the ladder, fence and wall, labelling the relevant distances. (b) Write equations that determine the minimum length of a ladder that will reach from the ground to the wall without being blocked by the fence. (c) Is a 20ft ladder long enough to reach from the to ground the wall without hitting the fence? Explain your answer. 5. (15 pts) A bar occupies the x-axis from x = 0 to x = L. The mass density of the bar at any point x is p(x) = x2 . (a) Determine the total mass of the bar. (b) Determine the location of the center of mass. (c) Sketch a graph of p(x) vs. x, and locate the center of mass on the x-axis. Is the center of mass closer to the left end or right end of the bar?
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6. (15 pts) Let V (a) be the volume obtained by rotating the region bounded by the coordinate axes, 1 the line x = a + a2 and the curve y = around the line x = -1. 1 + x6 (a) Give a formula for V (a). (Do not try to evaluate the integral!) dV (b) Evaluate . da (c) Is there a value of a > 0 for which V (a) = a? Explain your answer. 7. (10 pts) (a) Find an equation for the line tangent to the curve x3 + y 4 = 2 at (1, 1). (b...