Unformatted text preview: Math 191 FINAL EXAM Fall 2000 SHOW ALL WORK. CIRCLE YOUR ANSWERS. CLOSED BOOK. NO CALCULATORS. 1. (25 pts) Let f(:L‘) 2 m‘” 2 6“”, m > D (a) Compute lim f(:c) if it exists. 2—» 0+ (1)) Locate and identify the critical points of ﬂat) in a: > 0.
(c) Find the absolute maximum and absolute minimum of f (:3) in a: > 0 if they exist.
(d) Find the linearization of f at a: = 1. (e) Solve the equation mm = 1.01 approximately using one iteration of Newton’s method
(assuming that the initial guess is :30 = 1). 2. (10 pts)
Solve the initial value problem
i=(y2w4)sinx, y(0)=3 You may leave your answer as an implicit equation for the function y(3:). . (10 pts) 1 dt
(3.) Use the Trapezoid Rule with four subintervals to approximate the integral f 1 1 + 152' (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. . (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%ft high and 10§ft from the building. (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 ground to the well without hitting
the fence? Explain your answer. . (15 pts)
A bar occupies the sc—axis from a: = D to a: = L. The mass density of the bar at any point
a: is p(a;) : 2:2. (a) Determine the total mass of the bar. w (b) Determine the location of the center of mass. (0) Sketch a graph of p(x) vs. x, and locate the center of mass on the x—haxis. Is the
center of mass closer to the left end or right end of the bar? ...
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- Fall '07