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MATH118FS04N

MATH118FS04N - x = 0 for f x = e-x ln(1 x b ∞ ∑ n =0-1...

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MATH 118, Calculus 2, Final Exam, Spring 2004 Prob 1: Evaluate the following integrals. a) - 2 ln | x + 1 | + 3( x + 1) - 1 + tan - 1 ( x ) + ln( x 2 + 1) + C b) ln | x + 2 + x 2 + 4 x - 3 | + C Prob 2: Consider the parametric curve given by x = t 3 - 3 t - 2 and y = t 2 - 1 with - 2 t 2. a) Horizontal ( dy dx = 0 ) t = 0. Vertical ( dy dx = ) t = ± 1. b) Sketch the curve. c) t = ± 3. d) - 1 t 1. Prob 3: Consider the two polar curves r = cos θ and r = 1 - cos θ . a) Sketch the two curves. b) ( 1 2 , π 3 + 2 πk ) and ( 1 2 , 5 π 3 + 2 πk ) c) 7 π - 12 3 12 Prob 4: For each of the following series, determine whether it converges. Justify your answers. a) Diverges by the Comparison Test b) Converges by the Limit Ratio Test c) Converges by the Alternating Series Test 1

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2 Prob 5: a) Increasing and bounded between 1 and 9. It converges to 9. b) - 70 9 c) 4 9 Prob 6: a) Find the Taylor polynomial of degree 4 centered at
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Unformatted text preview: x = 0 for f ( x ) = e-x ln(1 + x ). b) ∞ ∑ n =0 (-1) n ( 1 x + 1 ) n Prob 7: Using the ±rst two terms we get 4 √ 83 ≈ 3 . 0185 with error < 1 200 by the alternating series test. Prob 8: Solve the following initial value problems. a) y = 1-2 x-1 / 2 b) y = 2 e x Prob 9: A cubical tank with sides of length 1 m is initially empty. Water enters from the top at a rate of 0 . 6 L/s = 6 10 , 000 m 3 /s and drains from a hole in the bottom of area 1 cm 2 = 1 10 , 000 m 2 at a speed of 4 √ y m/s , where y m is the depth of the water. Find the time taken for the tank to become full....
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