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Key points of this past exam of Calculus are: Fourth Degree, Indefinite Integral, Evaluate, Fourth Degree, Maclaurin Polynomial, Third Degree, Taylor Polynomial, Maximum Possible Error, According, Committed
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EXAM II - NOVEMBER 4, 2011
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
Total 100
1
2 EXAM II - NOVEMBER 4, 2011
1.(10 pts.)(a) Evaluate the indefinite integral ∫ ln(1 + x^2 ) dx.
(10 pts.)(b) Evaluate the indefinite integral ∫ x + 5 x^2 + 3x − 4
dx.
4 EXAM II - NOVEMBER 4, 2011
(10 pts.)(b) Let g(x) = √x^1 +2. Find the third-degree Taylor polynomial P 3 (x) for g(x) based at x 0 = 2.
MATH106B,C CALCULUS II - PROF. P. WONG 5
4.(10 pts.)(a) Let f (x) =
x. What is the maximum possible error, according to Taylor’s theorem, committed by using the third-degree Taylor polynomial P 3 (x) based at x 0 = 1 to estimate f (x) for 12 ≤ x ≤ 32?
(10 pts.)(b) Let
h(x) =
kx^3 , if 0 ≤ x ≤ 2; 0 , elsewhere.
Here, k is a constant. Determine the value(s) of k for which h(x) is a probability density function. Justify your answer.
