222-isaacs-07spex02 - CALCULUS 222 11th WEEK'EXAM I M...

Unformatted text preview: . CALCULUS 222 11th WEEK'EXAM I. M. Isaacs Thursday, April 12,2606/200? 5:30 — 7:00 P.M. Do all problems — 100 points. Use backs of pages for scrap, or if you need more space. NAME: TA: WW Do not write below here. Prob. l: I out of 25. _ Prob. 2: out. of 10. Prob. 3: out of 20. Prob. 4: out of 20. Prob. 5: out of 15. Prob. 6: out of 10. a“ Total: out of 100. V u 1. {25 POINTS] Find all of the following for the ellipse 25w2 + 1692 = 400. Length of major axis: Length of minor axis: Coordinates of foci: ( , —) and ( , ) Equations of directric-es: and Eccentricity: The total length of the path described below: Start at one focus, move up and to the right along a line having slope 1 until the ellipse is reached at some point P, and then and then move along a straight line from P to the other focus. A 1 + B cos 6' Find positive numbers A and B such that the new ellipse has exactly the same size and shape of the original ellipse. A: and B: Now a new ellipse is drawn. Its polar coordinate equation is 'r :2 Finally, find the Cartesian {that is 3:, y) coordinates of the foci of the new ellipse. Coordinates of foci: ( , ) and ( , ) [\‘J A. 2. {10 POINTS] Draw a reasonably accurate sketch of the graph of 43:2 — 8:6 — 9y2 + 36y = 68. Draw the asymptotes and mark the foci. (NEATNESS COUNTS ON THIS PROBLEM.) 3. {20 POINTS} (a) The graph of the polar coordinate equation 7” = 2(sin( 6) + 008(6)) is a circle. Find its cenmr and radius, 67mm polar coordinate equation is r = 2 cos(6). . 4, {20 POINTS] Let f (93): 1 + 2:3 ' (a) Compute the Maclaurin series of f(m) up to and including the $9 term. M I Compute the value of the ninth derivative of f at a: = 0. WM f . (50) — 562 (c) Compute i136 “Ts " ' . " T . . .n . 5. {15 pom-TS} Let f (cc) 2 sm Fmd the Taylor senes for the Iunctlon f (2:) expanded about the point a: = 1. (Write several terms, and then write the Whole series in Z-notation.) 6. [10 POINTS] I'have a function f that can be differentiated as many times as I want, and I can prove that for every number k 2 0, the kth derivative f (k) of f satisfies the inequality If“) < k! for all numbers x such that —2 < ac < 2. Let Pn be the nth Taylor polynomial for f (1') expanded around a: = 0. I want to use these Taylor~ polynomials to approximate the function f (a) I am sure that if I take 71 to be large enough, I can guarantee that the absolute value of the difference between Pn(.l) and f is very small. In other words, I know that lim Pn(.l) = f . n-—>oq Explain Why this is true. (b) Suppose I want to be sure that the absolute value of the difference between Pn(.1) and f is less than 10—6. How large shOuld I take 72? (0) Suppose that I want to approximate f (1.1) by Pn(1.1). Can I be sure that if n is very large, this will be a good approximation? Explain. THE END ...
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