Unformatted text preview: 10/03/2001 WED 17:03 FAX 6434330 MOFFITT LIBRARY I001
George M. Bergman Fall 1998, Math LAW 2 October. 1998 891 Evans Hall First Midterm, Makeup 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 ea or —-co. If none of these is true, write No limit. 2 3
5x + x
(a) lim _ — .
x .9 00 5x3 — x2
. ex — 1
(b) limx _> 0 2x
. x2 + 4
(C) llmx__> 2. fx—— 4 _ x .
(d) limx % 7: (sin x)/(x—1r). 2. (36 points, 9 points apiece) Compute the following derivatives. (Note that (c) is a
second derivative.) (a) iWLW
dx x3+2x2+79' (b) d—c:(secxb)a, where a and b are real numbers. d2 x
(C) m 6’" (d) d—c: f(x), where f is a differentiable function satisfying xf(x) —x2f(x) —- Xf(x)2 = 1. 3. (12 points) A point q is moving along the parabola y = x2. Express the rate of
change of its distance from (0,0) at a given moment in terms of x (the x—coordinate of
the point) and dx/dt (the rate of change of that coordinate). 4. (a) (8 points) Suppose f is a function and a a real number such that f is
differentiable at 0:. Give the deﬁnition of the derivative f ’(a). (b) (12 points) If f is a function and a a real number such that f is differentiable at
a, and ﬂat) i 0, prove from the above deﬁnition a formula for the derivative at a of
the function C(x) = 1/ f(x) in terms of f and its derivative at a. (You may assume
without proof results proved in Stewart about limits; and the result that a differentiable
function is continuous, but assume no differentiation formulas. In particular, you may not
assume the formula for the derivative of a quotient or for the derivative of a power; though
of course you may use either of those formulas in scratch—work to check the formula you get.) ...
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