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2007midterm1

# 2007midterm1 - dy dx = 3 x 2 cos(2 x with the initial value...

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MAT21B Midterm Dr. Qinglan Xia Oct.19, 2007 Time: 12:10–1:00pm Name Student ID Instructions. This is a closed book/notes/friends exam. Totally 4 pages, 6 problems. No calculator is allowed. 1. (10 points each) Evaluate the following integrals. Indicate the substitution that you make (if you made one). (a) cos x +1 x +1 dx (b) ln(2 x ) - 2 ln(2 x ) x dx (c) 200 x x 2 +1 ( 1+ x 2 +1 ) 101 dx 1

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2. (10 points each) Evaluate the following definite integrals. Indicate the substitution that you make (if you made one). (a) 1 0 e x +1 e 2 x dx (b) π 15 0 10 sec 2 (5 x ) tan(5 x ) dx . (c) 2 - 2 [sin ( x 3 + x ) + x 2 ] dx . 2
3. (10 points) Solve the differential equation

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Unformatted text preview: dy dx = 3 x 2 + cos(2 x ) with the initial value y (0) = 1. 4. (10 points)Find the area of the region enclosed by the curves y = sin 2 x + x 2 and y = x 2 over 0 ≤ x ≤ π . 3 5. (10 points)Using the sigma notation, express the upper sum and the lower sum of f ( x ) = x 2 9 + 1 obtained by dividing the interval [0 , 3] into n equal subintervals. Do not evaluate the sums. 6. (10 points)Calculate d dx Z e 2 x 3 x ln( t ) dt. 4...
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2007midterm1 - dy dx = 3 x 2 cos(2 x with the initial value...

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