MATH 224: INTEGRAL CALCULUS FINAL EXAM 2021

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MATH 224: INTEGRAL CALCULUS FINAL EXAM 2021 Instructions: Formulas: • Do not write above the horizontal line at the top of each page. • Put a ch... eck mark next to your section in the table to the left. • Read each problem carefully. • Write legibly and in complete sentences. • Cross off anything you do not wish graded. • This exam has 16 pages with 12 questions, for a total of 200 points. Make sure that all pages are included. • You may not use books, notes, or electronic devices. • You may ask proctors questions to clarify problems on the exam. • Graded work should only be written in the provided spaces (not on scratch papers) unless otherwise approved by a proctor. • You have 120 minutes to complete this exam. Good luck! sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos2 θ − sin2θ, sin2 θ = 1 (1 cos(2θ)), cos2 θ = 2 1 (1 + cos(2θ)), n . Σ n i = i=1 n(n + 1) , 2 Σn i2 i= 1 n(n + 1)(2n + 1) = , 6 Σ i3 = i= 1 n(n + 1) 2 2 . Math 224 1. For each statement below, if the statement is true, write T, and if the statement is false, write F, in the corresponding box. No explanation is necessary. (No partial credit is given on these problems.) Part (a) (b) (c) (d) (e) (f) (g) (h) T or F? (a) (5 points) If ˆ 5 f(x) dx = 12 and 1 ˆ 6 f(x) dx = 4, then 5 ˆ 6 f(x) dx = 16. 1 (b) (5 points) If lim . an+1 . = a 1, then the series Σ ∞ a cnonverges. n→∞ n n= (c) (5 points) If f J(x) > 0 on the interval [a, b], then Rn ≥ ˆ b n, where Rn is the right endpoint approximation. f (x) dx for every positive integer (d) (5 points) It is possible for a bounded non-monotonic sequence to diverge. ˆ M (e) (5 points) If f is continuous and ˆ ∞ f(x) dx converges to 0. f (x) dx = 0 for every M > 0, then the improper integral −M −∞ Σ∞ (f) (5 points) If n=1 Σ∞ (g) (5 points) If n= an is absolutely convergent, then lim an exists. n→∞ an is convergent, then lim |an| exists. n→∞ (h) (5 points) Some conditionally convergent series will converge, and others will diverge. Math 224 2. This problem has 2 parts. It concerns power series representation of functions. (a) (10 points) Begin with the power series centered at 0 of the function 1 . Then use the substitution t = 1 − x to represent the function 1 − t 1 as a power series centered at 1. x (b) (10 points) Represent the function ln x as a power series centered at 1. Math 224 3. This problem has 2 parts. It concerns Maclaurin series. (a) (5 points) Write down the Maclaurin series for the function cos x. No explanation is necessary. (b) (5 points) Find the exact value of the series . πΣ2 . πΣ4 . Σ6 . Σ8 6 π π 1 − + 2! 6 − 6 + 6 − · · · . 4! 6! 8! Math 224 4. (15 points) Find the Taylor series centered at 2 of the function f (x) = x3 − 5x2 + 6x + 8. Math 224 5. (10 points) Let a and b be fixed real numbers, where a = 0. Find the general solution to the differential equation yJ = ay + b. Math 224 x3 1 6. (15 points) Find the arc length of the curve y = 3 + 4x on 1 ≤ x ≤ 2. Math 224 7. (15 points) The base of a solid is a triangle with vertices at (0, 0), (0, 1) and (1, 0), and its vertical cross-sections perpendicular to the x-axis are all half-disks (i.e., filled-in semicircles). Find the volume of the solid. Math 224 8. The problem has 2 parts. It concerns the function f (x) = e−x2. d ˆ x (a) (5 points) Evaluate dx 0 f (t) dt, and justify your answer. (b) (5 points) Evaluate d ˆ 1 dx 0 f (x) dx, and justify your answer. Math 224 9. (15 points) Find the interval of convergence of the Taylor series Σ (x n + 3) . n=1 22n n Math 224 ˆ 10. (15 points) Find the indefinite integral 1 1 − x4 dx. Math 224 11. This problem has 2 parts. It concerns definite integrals. ˆ 2 (a) (10 points) Evaluate t4 ln t dt. 1 (b) (10 points) Evaluate ˆ 0 5 (x − 1)3 dx. Math 224 12. (15 points) Use trigonometric substitution to evaluate the following definite integral: ˆ 4 1 dx. 2√2 x2 x2 − 4 [Show More]

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