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 223Jr2233g + + 1
212233 22.2.333
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 ...
View
Full Document
 Spring '08
 WILKENING
 Math, Calculus, Moffitt Library

Click to edit the document details