**Unformatted text preview: **11/11/2001 SUN 14:01 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman £33] 1997, Math HlA 3 November, 1997
39 Evans Hall Second Midterm 1:10-2:00 PM 1. (40 points, 8 points apiece) Compute each of the following. A correct answer gives
full credit whether or not you show your computations. An incorrect answer, given with
computations that are correct except for a minor error, will get partial credit. (a) a 1n (ex+1). (b) f’(5), where f is the inverse function to g(x) = 2%3". 2 (c) The maximum and minimum values of the function f(x) = x + x‘ on the interval [1/2, 2], and the values of x at which these occur.
(d) 1imxg0 (ex _ 1)/sin x.
(e) The general antiderivative of sin(px + q), where p and q are constants with p¢0. 2. (30 points) Derive the formula d— sin—1 x = 1/ \/(x2 + 1). You may assume results dx
proved in the text before that formula. 3. (30 points) (a) (15 points) Suppose f is a continuous function on a closed interval
[(1, b], which is differentiable on the open interval (a, b), and that f ’(x) > 0 for all
points of that interval. Show that f is an increasing function on [a, b]. (This is
Stewart’s “Test for monotonic functions”. In your answer, you may use anything proved
in Stewart before that test.) (b) (10 points) Suppose again that f is a continuous function on a closed interval [(1, b],
but now let us only assume that for some cE(a, 1;), f is differentiable and has positive
derivatives on each of the open intervals (a, c) and (C, b) (but not necessarily at c).
Show that in this situation also, f must be an increasing function on all of [a, b]. In proving this you may use the result of (21) (even if you did not succeed in proving 6 it!), but you may not use the ‘generalized criterion for increasing functions” that I proved in class (of which this is a special case). (0) (5 points) Give an example of a function f satisfying the assumptions of (b), but not
all the assumptions of (a), for real numbers a, b, c which you should specify. You are not asked to prove that f has the properties required. ...

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