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Final Exam Preparation Sheet - Calculus II | MAT 152, Exams of Calculus

Material Type: Exam; Class: Calculus II; Subject: Mathematics; University: SUNY Institute of Technology at Utica-Rome; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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MAT 152, Calculus II
Final Exam Preparation Sheet
Problem 1. Find f(x) if
f00(x) = 1
2cosh(x/2) + 4
(2x+ 1)2, f0(0) = 0 ,and f(0) = 0 .
Problem 2. Find f0(x)
(a)f(x) = (cosh(5x))sin(4x)(b)f(x) = Zarctan(3x)
2 ln(x)qet+ tan2t dt
Problem 3. Evaluate the integral Rsin(5x)s in(2x)dx
Problem 4. (i) Sketch the region bounded by the curves y=xex/3,y= 0 and x= 0; x0.
(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=10
(x+3)(x2+1)
from x= 0 to x=3
4is rotated about the y-axis.
Problem 6. Find the length of the curve
y= 3 + x2
4ln 2x
2,1xe
Problem 7. Find the area of the surface obtained by rotating the curve x= 4+ y2
6,0y4
about the x-axis.
Problem 8. Find the exact length of the polar curve
r=e2θ,0θln 3 .
Problem 9. Determine whether the series is convergent or divergent. If it is convergent, find
its sum P
n=0 2nen+1
6n1
Problem 10. Use the Ratio Test to determine whether the series is absolutely convergent,
conditionally convergent, or divergent. P
n=1 (1)nn!5n
2n+1nn
Problem 11. Express the function as the sum of a power series by first using partial fractions.
Find the interval of convergence f(x) = 7
2x25x3
Problem 12. Evaluate the indefinite integral as an infinite series R3
2x3+ 3 dx
Bonus Problem 1. Find the volume of the resulting solid if the region under the curve y=
10
(x+3)(x2+1) from x= 0 to x=is rotated about the y-axis.
Bonus Problem 2. The Bessel function of order 0 is defined by
J0(x) =
X
n=0
(1)nx2n
n!(n)!22n
Show that J0satisfies the differential equation
x2J00
0(x) + xJ0
0(x) + x2J0(x) = 0
1

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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