midterm sample

# midterm sample - Math 1B Sample First Midterm π 1. (5...

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Unformatted text preview: Math 1B Sample First Midterm π 1. (5 points) Estimate the integral sin x using Simpson’s rule with n = 6. 0 2. 4 (6 points) Find 0 ∞ 3. (6 points) Find 0 4. ln x √ dx x arctan x dx (1 + x2 ) (π /2)2 − (arctan x)2 (6 points) Determine whether the series ﬁnd the sum. . Be careful to explain your steps. ∞ 3n + 5n converges or diverges. If it converges, 15n n=1 5. (7 points) Find the area of the surface obtained by rotating the curve y = ex , 0 ≤ x ≤ ln 2, about the x-axis. 6. (8 points) Find 5x2 − 5x + 5 dx. x3 − 2x2 + x − 2 7. (12 points) Find two of the following six integrals: [Caution: some of these integrals are impossible! ] (a). (b). (c). • • • • dx ln x √ sin x dx 3 x2 + 1 dx (d). sin2 x dx cos4 x (e). ex ln x dx (f). x x2 + 4x + 13 dx Be sure to circle the ones you want credit for. “Divergent” is an acceptable answer, but say why it diverges. “Impossible” is not an acceptable answer. Don’t forget the formulas on page 2. Some Formulas Trigonometric Identities 1. sin2 x = 1 (1 − cos 2x) 2 2. cos2 x = 1 (1 + cos 2x) 2 3. sin A cos B = 1 sin(A − B ) + sin(A + B ) 2 4. sin A sin B = 1 cos(A − B ) − cos(A + B ) 2 5. cos A cos B = 1 cos(A − B ) + cos(A + B ) 2 Trigonometric Integrals 6. u sin u du = sin u − u cos u + C 7. u cos u du = cos u + u sin u + C 8. un sin u du = −un cos u + n 9. un cos u du = un sin u − n 11. 12. 13. 14. un−1 sin u du a2 + u2 u a2 a2 + u2 du = a2 + u2 + ln |u + a2 + u2 | + C 2 2 1 u a2 + u2 du = (a2 + u2 )3/2 + C 3 u a4 u2 a2 + u2 du = (a2 + 2u2 ) a2 + u2 − ln |u + a2 + u2 | + C 8 8 √ √ a2 + u2 a + a2 + u2 du = a2 + u2 − a ln +C u u √ √ a2 + u2 a2 + u2 du = − + ln |u + a2 + u2 | + C u2 u Integrals Involving 10. √ un−1 cos u du ...
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## This note was uploaded on 02/19/2012 for the course MATH 1 taught by Professor Wilkening during the Spring '08 term at Berkeley.

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midterm sample - Math 1B Sample First Midterm π 1. (5...

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