calculus 2

Mathematics

University of Arizona

### Question Description

Attached you will find 14 Questions. Please, solve them with steps using calculus 2 methods. If the question is sequence or series, work all the test with steps.

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Math 156 Spring 2020 Final Exam – Part A This part has a total of 75 points, but is only worth 70 points. It is possible to get 5 points of extra credit. • Follow instructions for each problem carefully. • Submit each problem separately. • Show all of the steps. No credit will be given for answers without supporting work. • You must submit your own work. • You must use method covered in class this semester. If you want to use a different method, you must firs prove that this method works. • Make sure that your solutions are neat and your files are readable (not faint or blurry) and oriented correctly (not sideways). 1. (6 points) Set up an √ integral for the volume of the solid obtained by rotating the region bounded by the curves y = x, y = 1, and x = 5 about the line x = −2. You must sketch this region in order to receive any credit for this problem. Do not integrate. 2. (6 points) Sketch the region enclosed by the curves x = y 2 − 2y and y = x. Set up an integral for the volume of the solid obtained by rotating this region about the line x = 3. You must sketch this region in order to receive any credit for this problem. Do not integrate. Z 2 3. (6 points) Integrate p x2 1 + x3 dx. Your answer must be a fraction reduced to lowest terms. −1 dx Z 4. (6 points) Integrate (4 + x2 )3/2 Z 5. (6 points) Integrate 2x2 − 2x + 3 dx x3 + x 6. (6 points) Determine if the following series is absolutely convergent, conditionally convergent or divergent. In your response you must include: 1) The name of the correct test(s) that you used to reach the correct conclusion. 2) Work / reasoning to support your conclusion. 3) Your conclusion - either the series is (absolutely, conditionally) convergent or divergent. ∞ X (−1)n+1 n=0 4n + 3 5n3 − 2n 7. (6 points) Determine if the following series is absolutely convergent, conditionally convergent or divergent. In your response you must include: 1) The name of the correct test(s) that you used to reach the correct conclusion. 2) Work / reasoning to support your conclusion. 3) Your conclusion - either the series is (absolutely, conditionally) convergent or divergent. ∞ X 5 + 2 cos(4n) n=1 3n − 1 8. (7 points) Find the Taylor series representation for f (x) = cos(2x) centered at a = π/2. Use the definition of Taylor series. Your final answer should be in power series form. Determine the radius of convergence. 9. (7 points) Determine the center, the radius and the interval of convergence of the following series. ∞ X (−1)n−1 n=1 (ln n)n (x + 2)n (1 − 3n)n 10. (6 points) Consider the polar curve r = −6 cos θ. (a) Find the Cartesian equation for this curve. (b) Find polar coordinates (r, θ) for the point on this curve corresponding to θ = π . 4 (c) Find Cartesian coordinates (x, y) of the same point. (d) Sketch this curve and mark this point on the curve. 11. (3 points) Let X an be a convergent series. Evaluate the following limit. 2 − 5an 7an + 3 It must be clear how you obtained your answer. lim n→∞ 12. (4 points) Find a Cartesian equation of parametric curve x = 5 cos t, y = t3 + 1 by eliminating parameter t. 13. (3 points) Determine whether the following statement is true or false. If this statement is always true, write TRUE and provide an explanation. If this statement is false, write FALSE and provide a counterexample. Note: a counterexample is a specific example that shows that the statement is false. Your counterexample must be simple and obvious. If lim n→∞ q n |an | = 1 then the sequence {an } is convergent. 2 14. (3 points)Determine whether the following statement is true or false. If this statement is always true, write TRUE and provide an explanation. If this statement is false, write FALSE and provide a counterexample. Note: a counterexample is a specific example that shows that the statement is false. Your counterexample must be simple and obvious. If 0 < bn < an and the sequence {an } is convergent, then the sequence {bn } is also convergent. ...
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