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 3:104:00 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 xdirection is 1, and its velocity in the ydirection
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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