Practice Exam 2a

# Practice Exam 2a - y = √ 1 + e x , 0 ≤ x ≤ 1 about...

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Mathematics 115b Mid-term # 2 4/11/02 Name: Section: Please answer all EIGHT questions; each question is worth TEN points. Calculators, books and notes may not be used during the examination. 1 2 3 4 5 6 7 8 Total 1. Z sec 2 xdx = tan x + C 2. Z csc 2 xdx = - cot x + C 3. Z sec x tan xdx = sec x + C 4. Z csc x cot xdx = - csc x + C 5. Z sec xdx = ln | sec x + tan x | + C 6. Z csc xdx = ln | csc x - cot x | + C 7. Z tan xdx = ln | sec x | + C 8. Z cot xdx = ln | sin x | + C 9. Z sinh xdx = cosh x + C 10. Z cosh xdx = sinh x + C 11. Z dx x 2 + a 2 = 1 a tan - 1 ± x a ² + C 12. Z dx a 2 - x 2 = sin - 1 ± x a ² + C 13. Z dx x 2 - a 2 = 1 2 a ln ³ ³ ³ ³ x - a x + a ³ ³ ³ ³ + C 14. Z dx x 2 ± a 2 = ln | x + x 2 ± a 2 | + C

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(1) Evaluate (a) Z (ln x ) 2 dx . (b) Z cos 3 x sin x dx .
(2) Evaluate Z 2 1 x 2 - 1 x dx .

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(3) The velocity of a car was read from its speedometer at 15 minute intervals and recorded in the table below. Use Simpson’s Rule to approximate the distance the car traveled in 1 hour. time (minutes) velocity (miles/hour) 0 12 15 15 30 24 45 30 60 24
(4) Determine whether the integral is convergent or divergent. Evaluate it if convergent. (a) Z 0 1 cosh x dx . (b) Z 2 1 x 2 + x - 2 dx .

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(5) (a) Find the length of the curve y = ln( x 2 - 1), 2 x 3. (b) Find the area of the surface obtained by rotating the curve

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Unformatted text preview: y = √ 1 + e x , 0 ≤ x ≤ 1 about the x-axis. (6) Sketch the curve (an ellipse) given by parametric equations x = 3 cos θ , y = 2 sin θ (0 ≤ θ ≤ 2 π ) by plotting points. Indicate with an arrow the direction in which the curve is traced as θ increases from 0 to 2 π . Find the equation of the tangent to the ellipse at the point where θ = π/ 4. Calculate the area of the ellipse. (7) In each case, determine whether the sequence converges, and if so ﬁnd the limit. (a) a n = n + (-1) n n 2 n 2-n . (b) b n = n + (-1) n √ n √ n-n . (8) (a) Does the series ∞ X n =1 n n 2 + 1 converge or diverge? (b) Show that the series ∞ X n =0 (3-n-(-2)-n ) converges, and ﬁnd its sum....
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## This note was uploaded on 01/11/2010 for the course MATH 115 at Yale.

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Practice Exam 2a - y = √ 1 + e x , 0 ≤ x ≤ 1 about...

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