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Unformatted text preview: If! ‘I Name —....___ Instructor __....___— Class Time MATH 104 — Final EXAM
Wednesday, May 11, 2005, 1:30PM4z3OPM
McCosh Hall 46 This examination booklet contains 14 problems on 13 sheets of paper including the front cover. Do all of
your work in this booklet and show all your computations. This is a closed book exam. Calculators are NOT allowed. Your score WRITE OUT AND SIGN PLEDGE:
I pledge my honor that I have not violated the Honor Code during this examination. YOUR GRADED EXAM WILL BE OUTSIDE YOUR INSTRUCTOR’S OFFICE. DO NOT CALL
MATH OFFICE. ~/'2+51t2 +122 d: 1. (6' points) Find the value of the following integral and simplify your answer: jt‘ de 2 {6 points) Find the following integral: [:3 3. For each of the following integrals state whether it. converges or diverges, and give your reasons
carefully and clearly. (a) (5 points) [imam (.b) (5 mm) [a 273+. F:— 1 1 4. For each of the following series state whether it converges or diverges, and give your reasons carefully
and clearly. 3111f n3+n (a) {5 100an Sam n=1 (b) (5 pasneg :3 ”mm 5. (6 points) For which numbers p 2 0 does the following series converge? Justify your answer. 6. ( 5 points) What is the radius of convergence of the following power series? Justify your answer.
00 $1! 11:} i ' 7. (.9 points) For which a: does the following power series converge absolutely, converge conditionally, or
diverge? Justify your answer. a; Vlnn
5: a: 2
. ’ = "7 dt.
8 Let g(a:) [0 e
(a) {4 points) Find the ﬁrst four nonvanishing terms of the Meclaurin series of g(:c). 1 t
(b) (4 points) Estimate [0 3“"; dt with an error less than 0.01. Justify your answer. 003033)  1 + log(1 + at“) 9. (6 points) Find lime—+0 582(8’2 __ 1 _ £2) 10. (a) {3 points) Sketch the curve 1' = 1 — cos 9 given in polar coordinates on the following set of acres (on which the unit circle is already drawn). Shade the region that is inside the circle and outside
the curve you drew. (b) {5 points} Find the area of the region described above. '11. (7 points) Find all solutions of 24 + 227' + 4 = 0 in Cartesian form. 12. (6 points) Consider the area enclosed by the graph of y  e” , the line :r — 2, the line :c— * 3, and the
maxis. Set up an integral that expresses the volume of the solid obtained from rotating it. around
the axis x— — 5. (You do NOT need to calculate the integral. ) 13. (6 saints) Find all solutions to the difEerential equation . d.
awngi— = ﬂy. 14. A tank initially contains 30 liters of water, in which 100 grams of salt are dissolved. Brine that
contains 5 grams of salt per liter enters the tank at a. rate of 2 liters per minute. The solution is kept thoroughly mixed, and the mixed solution drains from the tank at the same rate of 2 liters per ,
minute. Let S(t) be the amount of salt in grams present in the tank after 2% minutes. (a) (3 points) Explain why S solves the initial valueproblem d_.S'(t) T =10———._S'(t), 5(0)=1oo. (b) (4 points) How much salt is in the tank after 1% minutes (exﬁressed in grams as a function of t)? ...
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 EdwardNelson
 Calculus

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