Individual Terms - Calculus - Exam, Exams of Calculus

Download Individual Terms - Calculus - Exam and more Exams Calculus in PDF only on Docsity! MATH 106 Final Exam Review, Part II 1. Find the solution to dy dx = cosx y2 that passes through (0, 2). 2. Use a second-degree Taylor polynomial to estimate 3 √ 28. 3. What is the largest possible error that could have occurred in your previous estimate? 4. Use a comparison to show whether each of the following converges or diverges. If an integral converges, give a good upper bound for its value. (a) ∫ ∞ 1 7 + 5 sinx x2 dx (b) ∫ ∞ 1 1 + 3x2 + 2x3 3 √ 10x12 + 17x10 dx 5. Decide if each of the following sequences {ak}∞k=1 converges or diverges. If a sequence con- verges, compute its limit. (a) ak = 3 + 1 10k (b) ak = (−1)k (c) ak = 3 + 5k 7 + 2k 6. Circle the appropriate word to complete each of the following statements correctly. (a) If the individual terms of a series approach 0 ( lim n→∞ an = 0), then the series ∞∑ n=1 an will converge (always/sometimes/never). (b) If the individual terms of a series approach 0.5 ( lim n→∞ an = 0.5), then the series ∞∑ n=1 an will converge (always/sometimes/never). (c) If the individual terms of an alternating series approach 0 ( lim n→∞ an = 0.5), then the series ∞∑ n=1 an will converge (always/sometimes/never). (d) If the individual terms of a geometric series approach 0 ( lim n→∞ an = 0), then the series ∞∑ n=1 an will converge (always/sometimes/never). (e) If the ratio of the terms of a series approaches 1 ( lim n→∞ an+1 an = 1), then the series ∞∑ n=1 an will converge (always/sometimes/never). (f) If the ratio of the terms of a series approaches 0.5 ( lim n→∞ an+1 an = 0.5), then the series ∞∑ n=1 an will converge (always/sometimes/never). (g) If a series has all positive terms, then it will converge to 0 (always/sometimes/never). 7. Decide if each of the following series converges or diverges. If a series converges, find its value. (a) 3.1 + 3.01 + 3.001 + 3.0001 + ... (b) 1 + 1/2 + 1/3 + 1/4 + ... (c) 5− 5/3 + 5/9− 5/27 + ...