Math 1B - Spring 1994 - Ribet - Midterm 1, 2, and Final

Math 1B - Spring 1994 - Ribet - Midterm 1, 2, and Final -...

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Unformatted text preview: 09/28/2000 THU 09:01 FAX 6434330 MOFFITT LIBRARY 001 Math 1B Spring, 1994 Professor K. Ribet First Midterm Examerbruary 22, 1994 ' 2 d 13. (6 points). Calculate / 1b (37 points Write a definite integral which represents the length of the curve y = tan m, nggw/él. 2 {7 points). Find the area of the surface obtained by rotating the curve 3,] = 81:30.17, 0 S a." g 271' about the line y : 0. Decide whether each of the following sequences converge or diverge. In the case of a convergent sequence, find the limit. Explain your reasoning! 3a {4 points). an = ——' . nsin(l/n) if n is odd 3b 13 . 6n = n s) ifn is even; 36 (4 points). cn=w+—-..+__ 1 4a {7’ points). Evaluate] msin_1wdm. 0 4b (7 points). Suppose that = 3. Use Simpson’s rule with n = 4 to estimate f(8): 3 2 2 4 5 (’7 points Write as a sum of partial fractions: 3:4 —i— 2.712 ' h ' ' m 2"” 2 d f 11 ' 1 6a (6 paints). Evaluate t e 1mproper 1ntegra1 1 $2 + 1 — w + 1 :c. (I t e rntegra is divergent, answer “divergent” and explain your reasoning.) ln 3 . d 6b (5 points). Find/1:12 gm i 1. Second midterm exam—April 7, 1994 00 1a points If the series 2(1 —— converges, calculate its value. If it is divergent, n=1 explain Why. 09/28/2000 THU 09:01 FAX 6434330 MOFFITT LIBRARY 002 sin a: , _ , (1:1: as an infinite series. 1 1b (5 points). Use Taylor series to express / 0 CC 251 (5 points}. Find the radius of convergence and the interval of convergence of the DC . . . (w — 2)” 1nfinlte serles E —3—~—. n 71:1 00 2b (4 points). For What positive values of .1: does the series + converge? n n=1 3 . . . 1 . . 3 {7 pomts). Show that Ismx — a: + < 10—7 if 0 < a: < [Coriander the Maclaurm polynomial of degree 4 for sin 4a (6 points). Find y(t), given that 7531' + y = — sint and that Mar/2) = 0. x—y 4b (5 points). Solve the homogeneous differential equation 3/ = s Decide Whether each of the following series converges or diverges. Explain your reasoning! . m n . 1 5a (4130212253). 25—1) 718111;; 00 n! 51) (4 points). 2 —; n71; n=4 ' 0° (InnJM! t . ‘ 5c (4 pom 3) “2:31 TIN/E , l l l l 1 5d ' _ { g A. , , _ (Moms) 2 2.3 2-2-3Jr2-2-3-3g + + 1 212-2-3-3 2-2.2.3-3-3 1 What is the radius of il- 6 (7 points). Find the Maclaurin series Jfor 3 8 — w convergence of this series? Final exam “May 2],, 1994 (—1)”. nlfn ' 00 1b {6 points]. The alternating series 2 n=1 a. Converges conditionally; _ b. Converges absolutely; w C. Diverges; % +1 1 __ 1c (’7 points). Evaluate the improper integral / 1 f' 1 + 3: dx. (If the integral is divep _1 33 gent, answer “divergent” and explain your reasoning.) 2 09/28/2000 THU 09:01 FAX 6434330 MUFFITT LIBRARY ‘003 2a (5 points). For which values of a: does the series Z + (312)”4'1) converge? Find n20 the sum of the series for those values. (LN 21) (7 points]. Consider the differential equation —.— : (N + 1)(N — 2)(N — 3)(N a 4). oft Suppose that NOE) is a solution to the equation with N({]) z 1.5. Then tlim NU): —>oo . Is necessarily —1 h; Is necessarily 2 ; Is necessarily a number other than —1 or 2 Can be any real number ; 5°99?” Does not exist 3a {6' points). Find /(cos:r + sin 3:)2 tanxdw. 3b {5 points}. Given = 0, use Simpson’s rule with n : 4 to estimate f(8): 2 l 4 (9 points Calculate the integral / (is. Check your work carefully! 5a (6 points). Find lim {72” +4”. 71—400 5].) (5 points). Find the equation of the curve which paSses through (1,1) and Whose slope at each point (:6, y) on the curve is y2/5L'3. 6a (’7 points). Find f(m)([)) Where = (1 + ac) cos :c. m 613 {6 points Determine Whether the sum 2 n n=1 7 (8 points). A tank contains 25 kg of salt dissolved in 500 L of water. Brine that contains 0.03 kg of salt per liter of Water enters the tank at the rate of 25 L/hour. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the mixture after 41' hours? 8a {6' points). Find given: f”(zr;) — 2f’(ac) +2f($) = 0, : l, f’(0) = 1. Sb (7 points). Solve the difi'crential equation y” + y 2 w + cos 3:. 9 {8 points). Find a series solution for y” = 2.233;" +23; with the initial conditions y(0) = 17 I y (0) = 0- ...
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This note was uploaded on 04/02/2008 for the course MATH 1 taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

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