Math 1A - Fall 1997 - Bergman - Midterm 1

Math 1A - Fall 1997 - Bergman - Midterm 1 - SUN 13:48 FAX...

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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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