Calculus II Final Exam Solutions

A/ Integrate the following  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) (30)

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B/ Make a diagram for questions 1 - 4.

1) Points of intersection: (-1, 5) and (3, -3) (5)

2)

 R is y = x2 + 2 r is y = ½ x + 1 (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!)

 1)      so, y = eº = º .       .  We apply l'Hopital's rule to ln y we get ln y = 1 therefore y = e.

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(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.

 a) GS: a = 2, r = 1/3,converges, Sum = 3 b) p-series: p > 1converges, Sum = 3/2 c) Diverges

(8)

2) a) Function must be continuous, positive, and decreasing.

(3)

b) The integral from 1 to º is ln x so diverges to º

(3)

TOTAL (100)

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