**Unformatted text preview: **11/11/2001 SUN 13:48 FAX 6434330 MOFFITT LIBRARY I001
George M. Bergman Spring 1997, Math HIA 26 September, 1997
39 Evans Hall First 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) limx _> 2 (x3 — 8)/(x2 — 4). (Give a real number, or +00, or woo, or say 77 “Undeﬁned & not +00 or —oo
(b) limx _) Fm (x3 — 8)/(x2 — 4). (Same choices as for (a).) 5 (c) “SF-“xx sin x. 2
(d) “3—3 f(1/x), where f is twice differentiable.
x (e) The equation of the line tangent to the curve x2 — y2 = 5 at the point (3, 2). 2. (10 points) Complete the following precise deﬁnition: Let f be a function deﬁned on the real line, and L a real number. Then limx _> “00 f(x) = L means that 3. (20 points) For F (x) = f(x)g(x), derive using the deﬁnition of derivative the formula
F’(a) = ﬂu) g’(a) + f’(a)g(a), where f and g are differentiable at a. You may use
facts we have proved about limits, but not further facts proved about derivatives (such as the above formula). 4. (30 points) Suppose p and q are polynomials, and n apositive integer. Prove that n
d n p(x)/q(x) can be written with denominator 90:)" +1', i.e., that there is a polynomial
3C
61" ._ n+1
a(x) such that d n p(x)/q(x) — a(x)/q(x) .
x You may take for granted that a sum, product, or difference of polynomials is a
polynomial, and that the derivative of a polynomial is a polynomial, and thus that any
expression obtained from polynomials by any combination of these operations is a
polynomial. You may also assume any results proved in the course so far. Suggestion:
'Use mathematical induction. (You will get partial credit for merely setting up the
induction, i.e., for writing down what has to be proved to give such a proof. Once you
write this down, the result will be a fairly straightforward computation. If you can’t give a
full proof, you can get some further partial credit for verifying the result for n. z 1, 2, 3.) ...

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