NAME:
Math 106C - Final Exam, December 14, 2006
INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work.
1. (9 points) Solve the IVP y0=sin2(t) cos3(t)
y,y(0) = 4.
2. (7 points) Evaluate Z5
2
3
x−4dx.
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Key points of this exam are: Arclength, Circle, Unnecessary Work, Evaluate, Integral, Curve, Point, Plane, Graphs, Rotating
Typology: Exams
2012/2013
1 / 7
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NAME:
Math 106C - Final Exam, December 14, 2006
INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work.
- (9 points) Solve the IVP y′^ =
sin^2 (t) cos^3 (t) y , y(0) = 4.
- (7 points) Evaluate
2
x − 4
dx.
- (7 points) Evaluate
0
xe−^2 x^ dx.
- (5 points) Write an integral that gives the arclength of the curve y = x^2 + 1 from the point (1, 2) to the point (2, 5). (You do not have to evaluate the integral.)
- (7 points each) For each of the following, determine whether the series converges or diverges, and state the test that you are using. If the series converges, find an exact value or give upper and lower bounds.
(a) 12 − 6 + 3 − 32 + 34 + · · ·.
(b)
∑^ ∞
k=
k^2 + 1
(c)
∑^ ∞
k=
2 k + 3 4 k + 5
(d)
∑^ ∞
k=
arctan (k) − arctan (k + 1). (Hint: Write out the first few partial sums.)
- (10 points) Find the radius and interval of convergence of the power series p(x) =
∑^ ∞
k=
(x + 3)k k 5 k^
- (5 points) What is the alternating harmonic series? Does it converge absolutely or conditionally? Explain how you know.