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Unformatted text preview: Final Examination — MATH 141 —— May 15, 2008 Instructions: Answer each of the 9 numbered problems on a separate answer sheet. Each answer
sheet must have your name, your TA’s name, and the problem number (=page number). Show all your
work for each problem clearly on the answer sheet for that problem. You must show enough written work
to justify your answers. Finally, please copy and sign the Honor Pledge on page 1. NO CALCULATORS OR OTHER ELECTRONIC DEVICES 1. (a) (10) Calculate the volume V of the solid generated by revolving around the 3: axis the bounded
region between the graphs of f = sins: and g(a:) = cosar: on the interval [0, 7r (b) (10) Let the curve C be given parametrically by :1: = 1 — t2 and y = 1 + t3 for 0 S t S 1. Find
the length L of C. ‘ 2. (a) (10) Suppose the work done in extending a spring from its natural length to 4 inches extended
is % ftlb. Find the work W done in extending the spring 2 more inches. (b) (15) Consider the bounded region R between the graphs of f and 9, where f = 3x and
g(:r) = $62. Sketch R,'with appropriate labels. Then set up all of the integrals that are required
in order to determine the center of gravity of R. DO NOT EVALUATE THE INTEGRALS. “:3 4—3: (b) (10) Consider the differential equation y\/ x2 + ldy = xdx. Find the unique solution of the
differential equation that satisﬁes = 4. 3. (a) (10) Let ﬂit) = , for a: < 4. Show that f has an inverse, and ﬁnd (f‘1)’(1/3). 4. (a) p (5) Simplify the expression tan (sec‘1 032).
(b) (15) Let f(x) = 332$“. Find f’(a:) and lim m—>O+ 5. (a) (10) Evaluate /———1————dw. w2 102—9 (b) (10) Evaluate / t sec2tdt. 00
6. (a) (10) Determine if / $2 e2m3 da: is convergent, or divergent. If convergent, ﬁnd its value.
1 (b) (10) Evaluate (n —— 2)1/n, giving reasons. 7. (a) (10) Compute lim 9:2 tan(1/m3), giving reasons.
w—+oo °° 71.2k+1
(b) (10) Find the numerical value of Z 3k_1 .
k=2
°° 1
(c) (10) Determine Whether the series 23—1),“ [can k)2 converges absolutely, converges condition—
[6:2
ally, or diverges. Give reasons.
00
8. Let ﬂux) = 20—1)" 3“ 1102"“.
n=l
(a) (10) Find the radius of convergence R of f.
1/3
(b) (10) Use the given power series to ﬁnd an approximation of f (t) dt that has an error of 0
less than 0.001. Don’t simplify your answer. 9. (a) Let z = —8. Write each of the 3rd roots 21, 22, 23 of z in the form rem. Then display these
roots in the complex plane, and identify each of them. (b) (15) Consider the cardioid deﬁned by r = 2 —— 2 sin 0. Sketch the cardioid in the plane (including
pertinent labeling), and ﬁnd the area A it encloses. END OF EXAM —— GOOD LUCK! ...
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