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Unformatted text preview: MATH 104  FINAL EXAM
Wednesday, May 14, 2003, 1:30PM4z30PM
McCosh 50 ‘1. (10 points) Find the following integrals. (a) fezzsin(e‘”)da: (b) AIUx/éz—Trﬁdm _2. (12 points) (a) Let R be the region bounded by the curve y = 3:3, the x—axis and the two vertical lines a: = 1
and m = 2. Find the volume of the region obtained by rotating R about the line at = 3. (b) Let C be the portion of the curve y— — :03 between the points (1,1) and (2, 8). Find the area of
the surface generated by rotating 0 about the 3:— axis. ‘3. (15 points) Determine whether the given improper integrals converge or diverge. Justify your answers.
\ (a) [:0 sin2(a:) d2; :1:(ln 3:)? (b) [01 sin(a:2) d3: 35/2 °°arctan(:c 2
(c) / $)d Note: arctan(:c2) =tan'1(x2) z3+\/Ed 1 3/2
d “5‘
( ) [o In(1+x2) d3; (e)/1md xii—1 4. ( 15 points) Write AC or CC or D to indicate whether the given series is Absolutely Convergent,
or Conditionally Convergent or Divergent. Justify your answers. oo (_1)n . 5. (10 points) Let 0 g 6' S 27r and consider the series Z(—1)"+1 tan2"(0). Determine the values of 0
:1
for which the series converges and compute the sum. "Simplify your answer. .6. {.9 points) Find the ﬁrst three nonzero terms of the Taylor series at 0 for the function f (x) = $2
+ x
. . . (ee’ — 1 — 2x2)(cos(x) — 1)
‘7~ (1 ‘7 Pom“) Fm“ £13?) [sin(3z) — ln(1 + 325W
‘ 8. (9 paints) Let z = ~5— + ~2— z and let 10 be the complex number whose modulus is 2 and whose . argument is 7r/ 3. (Note: The modulus of a complex number is the same as the magnitude.) Write
each of the quantities below inthe form a + ib where a and b are real numbers. (8!) ~ .2 (b) 280 (c) 2211; 9. (10 points) Find all complex numbers 2 satisfying the equation (22 — 1)4 = —16. Express your
answers in the form a + ib, where a and b are real. ...
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