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Material Type: Exam; Class: Calculus II; Subject: Mathematics; University: SUNY Institute of Technology at Utica-Rome; Term: Unknown 1989;
Typology: Exams
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MAT 152, Calculus II Final Exam Preparation Sheet
Problem 1. Find f (x) if
f ′′(x) =^12 cosh(x/2) + (^) (2x 4 + 1) 2 , f ′(0) = 0 , and f (0) = 0.
Problem 2. Find f ′(x)
(a) f (x) = (cosh(5x))sin(4x)^ (b) f (x) =
∫ (^) arctan(3x) 2 ln(x)
√ et^ + tan^2 t dt
Problem 3. Evaluate the integral ∫^ sin(5x) sin(2x) dx Problem 4. (i) Sketch the region bounded by the curves y = xe−x/^3 , y = 0 and x = 0; x ≥ 0. (ii) Find the volume generated by rotating the region about the y-axis. Problem 5. Find the volume of the resulting solid if the region under the curve y = (^) (x+3)(^10 x (^2) +1) from x = 0 to x = 34 is rotated about the y-axis. Problem 6. Find the length of the curve y = 3 + x 42 − ln 2 2 x, 1 ≤ x ≤ √e
Problem 7. Find the area of the surface obtained by rotating the curve x = 4+ y 62 , 0 ≤ y ≤ 4 about the x-axis. Problem 8. Find the exact length of the polar curve r = e^2 θ^ , 0 ≤ θ ≤ ln 3. Problem 9. Determine whether the series is convergent or divergent. If it is convergent, find its sum ∑∞ n=0 2 n 6 nen−+1 1 Problem 10. Use the Ratio Test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑∞ n=1 (−1)n^2 nn+1!5nnn Problem 11. Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence f (x) = (^2) x (^2) −^75 x− 3
Problem 12. Evaluate the indefinite integral as an infinite series ∫^3
2 x^3 + 3 dx Bonus Problem 1. Find the volume of the resulting solid if the region under the curve y = (x+3)(^10 x^2 +1) from^ x^ = 0 to^ x^ =^ ∞^ is rotated about the^ y-axis. Bonus Problem 2. The Bessel function of order 0 is defined by
J 0 (x) = ∑^ ∞ n=
(−1)nx^2 n n!(n)!2^2 n
Show that J 0 satisfies the differential equation
x^2 J 0 ′′ (x) + xJ 0 ′(x) + x^2 J 0 (x) = 0