Math 1A - Fall 1998 - Bergman - Midterm 1

# Math 1A - Fall 1998 - Bergman - Midterm 1 - 1

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Unformatted text preview: 10/03/2001 WED 16:27 FAX 6434330 MOFFITT LIBRARY I001 George M. Bergman Fall 1998, Math 1AW 2 October, 1998 2050 VLSB First Midterm Exam 3:10-4:00 PM 1. (32 points, 8 points apiece) Compute the following limits. Give the value if the limit is deﬁned, or if it is so or —oo. If none of these is true, write NO limit. 3 5+x (a) lim . xeaa 5—x2 3x . e —1 (b) 11mx__§0 x 2 (6) lim x ‘4 x~—>2 Ji—m‘ (d) limx sin x. ——)00 2. (36 points, 9 points apiece) Compute the following derivatives. (Note that (c) is a second derivative.) 1 5 4 (a) dx(x +23: +79) (b) d—dx(cosx“)b, where a and b are real numbers. d2); (C) We . (d) 2% g(x), where g is the inverse of the function f(x) = x3 + x. 3. (12 points) A point p is moving in the (x, y) —plane (happily unaware that three hundred and forty—four Math 1A students are thinking about it). At a certain moment, its position is (x0, yo), its velocity in the x-direction is 1, and its velocity in the y-direction is 2. Find the rate of change of its distance from (0,0) (in terms of x0 and 3’0)- 4. (a) (8 points) Suppose f is a function and a a real number such that f is differentiable at a. Give the deﬁnition of the derivative f ’(a). (b) (12 points) If f and g are functions, and a: a real number such that f and g are differentiable at a, prove from the above deﬁnition a formula for the derivative at a of the function H(x) = f(x) — g(x). (You may assume without proof results proved in Stewart about limits; but assume nothing about derivatives except the deﬁnition.) ...
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