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Unformatted text preview: Math 164 Final Exam, Fall 2008
1. Find y ′ if (a) y = 2x ln 4x (b) y = arctan ex
2 2. Find lim
x→∞ 1− 1 1 + 2 x x x 3. Evaluate the following integrals. (a) x ln xdx (b) x2 dx (x2 + 4)3 (c) x2 + 1 dx x(x2 + 4) 4. Evaluate the integral.
1 0 √ dx 1−x 5. Find an equation of the line which is tangent to the parametric curve x = 2 cos t, y = sin t, √ √ with 0 ≤ t ≤ π , at the point ( 2, 22 ). 2 6. Find the length of the arc of the curve y 2 = 2x3 from point (0, 0) to point (2, 4). 7. Set up, but do not evaluate, an integral which represents the area of the region which lies between the curves: r = 1 + sin θ r = 1 − sin θ 8. Determine the limit of the following sequence, if it exists. Make sure and show all work involved in the calculation. 1 − 3n 4n + 1
∞ n=1 9. Determine whether or not the following series is conditionally convergent, absolutely convergent, or divergent. State carefully which test you are using.
∞ n=2 (−1)n n2 n +1 10. Determine whether or not the following series converges or diverges. State carefully which test you are using.
∞ n=1 √ n3 1 +n+1 11. Find the radius of convergence of the power series
∞ n=1 n2 (x + 7)n+1 10n 12. Find the Maclaurin series for 2x 1 + x2 Bonus Determine whether or not the following series converges or diverges. If it converges, ﬁnd the sum of the series.
∞ n=0 2n n! ...
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This note was uploaded on 11/04/2010 for the course MATH 164 taught by Professor Staff during the Spring '08 term at IUPUI.
 Spring '08
 STAFF
 Integrals

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