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The final exam for a calculus ii course taught by professor p. Wong on april 7, 2006. The exam covers various topics including indefinite integrals, improper integrals, volume of solids of revolution, initial value problems, and series. Students are required to explain their work and provide reasons for their answers.
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FINAL EXAM - APRIL 7, 2006
Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1
2 FINAL EXAM - APRIL 7, 2006
(9 pts.)(b) Evaluate the following improper integral. ∫ (^) ∞ 0
e−^2 x(x + 1) dx
4 FINAL EXAM - APRIL 7, 2006
(20 pts.)3. Find the volume of the solid formed when the region bounded by the curve y = arctan x, the axis y = 0, and the line x = 1 is revolved around the y-axis. [First sketch a picture of the region.]
MATH106B CALCULUS II - PROF. P. WONG 5
2 x ,^ y(1) = 0. (9 pts.)(a) Use the method of separation of variables to solve this IVP.
(9 pts.)(b) Estimate the value y(2) (when x = 2) of the solution using Euler’s method with two steps with initial point (1, 0). DO THIS BY HAND.
MATH106B CALCULUS II - PROF. P. WONG 7
(−1)n 5 nn! (2n)!
(9 pts.)(b) (^) ∞ ∑ n=
(−1)n n + 3
8 FINAL EXAM - APRIL 7, 2006
(10 pts.)(a) Let f (x) = ecos^ x. Find the second degree Maclaurin polyno- mial for f (x).
(10 pts.)(b) Use the Maclaurin series representation for (^1) −^1 x to find the Maclaurin series for g(t) = (^) 1+^1 t 3.
