MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS

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1 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS 1. We have one theorem whose conclusion says an alternating series converges. We have another theorem whose conclusion says an alternating series diverges. Substitute b n = into these theorems. Is the following 2n 2 +8 series convergent of is it divergent? Justify your answer. 2n The Alternating Series Theorem says that for the alternating series 2n ; if (1) lim n 2n 2 +8 = 0 and (2) 3n+10 2(n+1) 2 +8 < 2n 2 +8, then 2n converges. The Divergence Theorem for Alternating Series says that for the alternating series 2n ; if lim n = L and L 0, then 2n n diverges. The series is convergent: lim n 2n 2 +8 = lim n 3/n+7/n2 = 0. So, (1) 2+8/n 2 holds. 3n (n + 1) < 3n + 7 2n 2 (3n + 10)(2n 2 + 8) < (3n + 7)(2(n + 1) 2 + 8) + 8 6n n n + 80 < (3n + 7)(2n 2 + 4n + 10) 6n n n + 80 < 6n n n < 9n n Since 10 < 9n n is clearly true for n 1, we conclude that (2) is true. Since (1) and (2) are true, we conclude that 2n converges. 2. Is the following series convergent or divergent? Start by substituting b n = 2n+8 into one of the theorems about alternating series. Justify your answer to the question using this statement. 2n + 8

2 2 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS The Divergence Theorem for Alternating Series says that if 2n+8 = L and L 0, then lim n lim n 2n+8 = lim n 3+7/n 2+8/n = So, we conclude that 3. Is the following series convergent or divergent? Start by substituting b n = 3n2 +7 2n+8 into one of the theorems about alternating series. Justify your answer to the question using this statement. n 1 3n n + 8 3n A theorem says that if lim 2 +7 n 2n+8 does not exist, then n 1 3n n lim 2 +7 n 2n+8 = lim n 3+7/n2 which does not exist. 2/n+8/n 2 So, we conclude that n 1 3n Find the value of L in the ratio test for the following series. Write a (2n + 1)x n 3 n (5n 2) L = lim (2n + 3)x n+1 n 2 n+1 (5n + 3) 3n (5n 2) (2n + 1)x n = lim x(2n + 3)(5n 2) n 3(2n + 1)(5n + 3) = lim x(10n 2 + 4n 6) n 3(10n 2 + 8n + 3) = x 3 lim /n 6/n 2 n /n + 3/n 2 = x 3 The ratio test says that if 0 x 3 < 1, then (2n + 1)x n 3 n (5n 2) converges.

3 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS 3 So, we conclude that the series converges for x such that x < Find the value of L in the ratio test for the following series. Write a n (2n + 1)x2n 3 n (5n 2) L = x 2 /3. The series converges for x such that x < Find the value of L in the ratio test for the following series. Write a (2n + 1)x n 3 n n! L = 0. The series converges for all x. 7. Find the value of L in the ratio test for the following series. Write a (2n + 1)x 2n 3 n (2n)! L = 0. The series converges for all x. 8. Find the value of L in the ratio test for the following series. Write a n!x n 3 n (5n 2) L does not exist if x 0. L = 0 if x = 0. The series converges only for x = 0. 2n 6 9. Find constants c and p such that n(n 2 2) < for all n 1. Show that the following series is convergent using the comparison test. 2n 6 n(n 2 2) c = 3, p = 2. Other values of c and p would also work. c n p

4 4 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS 2n Find constants c and p such that n(n 2) > c n for all n 2. Show p that the following series is divergent using the comparison test. 2n 6 n(n 2) c = 1, p = 1. Other values of c and p would also work. 11. You can assume that the following alternating series satisfies the hypotheses of the alternating series test. If x 0.4 find the smallest value of N such that k 1 x 2k k(k + 1)! < k=1 N = Suppose we use N+1 N k=1 k 1 to approximate the sum of the series (k 2) 3 k 1. What is the smallest value of N for which we can be sure that (k 2) 3 the error is less than 10 4? N = Assume it is known that the following series satisfies the hypotheses of the alternating series theorem for all x. If x 0.5 and N = 4 find a bound for k xk 2k! k=n Find the Taylor polynomial T 4 (x) about a = 0 for the function f(x) = e 2x + (x + 9) 1/ x x x x4 15. The Taylor polynomial T 4 (x) for (8 + 3x) 2/3 is (8 + 3x) 2/3 4 + x 1 16 x x x4. (a) Using differentiation find the Taylor polynomial for (8 + 3x) 1/3. (b) Using integraion find the Taylor polynomial for (8 + 3x) 5/3.

5 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS 5 (a) x x2 7 (b) x x x x x x4 16. The Taylor polynomial T 4 (x) for (1 + 2x) 1/2 is Find T 4 (x) for (9 + 2x) 1/2. (1 + 2x) 1/2 1 x x2 5 2 x x x x x x4 17. The power series for sin x converges for x < 1. For what range of values of x is x3 sin x [x 3! + x5 5! ] < 10 7? x < Write the power series (3n 2)x n n=3 (3n + 7)x n+3 in standard position. 19. The graphs of r(t) and R(s) are lines. Find the point where these lines intersect. (1, 4, 3) r(t) = (3t 2) i + ( 2t + 6) j + (7t 4) k R(s) = (2s + 2) i + (4s + 6) j + ( 2s + 2) k 20. Consider the graphs of the following two vector functions. Show that the two graphs are actually the same line. r(t) = ( 5t + 2) i + (2t 3) j + (t + 1) k R(s) = (10s 23) i + ( 4s + 7) j + ( 2s + 6) k r(0) = (2, 3, 1) = R(5/2) and r(1) = ( 3, 1, 2) = R(2). Since the two lines share 2 points, the lines must be the same line. 21. Let r(t) = (3t 2) i + (2t) j + ( 3t + 1) k. (a) Find a line parallel to r(t) that contains the point (1,1,2).

6 6 MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS (b) Find a line perpendicular to r(t) that contains the point (1,1,2). (a) R(t) = (3t + 1) i + (2t + 1) j + ( 3t + 2) k (b) R(t) = (2t + 1) i + ( 3t + 1) j + 2 k 22. Find a vector perpendicular to the plane containing the three points (2,1,3), (0,-2,1), and (-1,2,5). 4 i + 10 j 11 k 23. Let v = 2 i k and w = i + 3 j 2 k. (a) Find the angle between the vectors v and w. (b) Find P roj w v. (c) Find P roj v w. (a) (b) 2 7 ( i + 3 j 2 k) (c) 4 5 (2 i k) 24. Find the arc length of the curve r(t) = 4t i 2 2t 2 j +( 4 3 t3 4) k from the point where t = 1 to the point where t = Find the area of the parallelogram with vertices (1, 2, 3), (4, 5, 2), (2, 2, 2), and (3, 1, 3). 38

AP Calculus Testbank (Chapter 9) (Mr. Surowski)

AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series