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Some past exams of Calculus for students. Keywords of the exam are: Value, Definite Integral, Indefinite Integral, Function, Satisfies, Third Degree, Taylor Polynomial, Possible Error, Committed, Maclaurin Polynomial
Typology: Exams
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EXAM II - MARCH 9, 2007
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 EXAM II - MARCH 9, 2007
1.(10 pts.)(a) Find the exact value of the definite integral ∫ (^2) 1
x ln(x^2 ) dx.
(10 pts.)(b) Evaluate the indefinite integral ∫ (^2) x (x + 1)(3x − 1) dx.
4 EXAM II - MARCH 9, 2007
(10 pts.)(a) Write down the third-degree Taylor polynomial P 3 (x) for f based at x 0 = 1.
(5pts.)(b) Suppose it is known that for 0 ≤ x ≤ 2, |f (4)(x)| ≤ 0 .5. What is the maximum possible error committed by using P 3 (x) to estimate f (x) for 0 ≤ x ≤ 2?
(5 pts.)(c) Suppose g(x) = f (x + 1). Use part (a) to find the third-degree Maclaurin polynomial for g.
MATH106A CALCULUS II - PROF. P. WONG 5 4.(10 pts.)(a) Let f (x) = cos(2x). Find the fourth-degree Maclaurin polynomial M 4 (x) for f.
(10 pts.)(b) Solve the following Initial Value Problem y′^ = (1 + x^2 )ey, y(0) = 0.
