Calculus II Final Exam Solutions |
A/ Integrate the following
(30)
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B/ Make a diagram for questions 1 - 4.
1) Points of intersection: (-1, 5) and (3, -3)
(5)
2)
(5)
3) y = 2x - x² = x(2 - x) = 0 at x = 0 and x = 2 .
(5)
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4) base = 2y, therefore A = 2y² = 2(16 - x²)
(5)
5) (5)
6)° f(x) = 27 = 3(3z²), so z = 31/ 2 .
(4)
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C/ Find the limits (Show all your work!)
We apply l'Hopital's rule to ln y we get ln y = 1 therefore y = e. .
(15)
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D/ Evaluate if possible
(12)
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E/ Test for Convergence
1) Test these infinite series. If it converges, find the sum. State the test used.
converges, Sum = 3 converges, Sum = 3/2 (8)
2) a) Function must be continuous, positive, and decreasing.
(3)
b) The integral from 1 to º is ln x so diverges to
3) ln| x | - ln| x + 3 | + 2 ln| x - 1 | + C
4) ln | x² - 6x + 13 | + 5/2arcTan ½(x - 3) + C
5) 2x1/2 - 3x1/3 + 6x1/6 - ln | x1/6 + 1 | + C
6)
R is y = x2 + 2
r is y = ½ x + 1
1)
so, y = eº = º
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a) GS: a = 2, r = 1/3,
b) p-series: p > 1
c) Diverges
(3)
TOTAL (100)
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