math111_2005fall_prelim3sol - Math 111 Prelim 3 November...

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Unformatted text preview: Math 111 Prelim 3 November 29, 2005 Name: Instructor: Section: INSTRUCTIONS READ THIS NOW This test has 10 problems on 10 pages worth a total of 100 points. Look over your test package right now . If you find any missing pages or problems please ask a proctor for another test booklet. Write your name, your instructors name, and your section number right now . Show your work. To receive full credit, your answers must be neatly written and logically organized. If you need more space, write on the back side of the preceding sheet, but be sure to label your work clearly. This is a 90 minute test. You are allowed to use your calculator and a 3 5 inch index card of notes. All other aids are prohibited. You DO NOT need to SIMPLIFY your answers. OFFICIAL USE ONLY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Total: Math 111 (Fall 2005) Prelim 3 (11/29/2005) 2 1. [16pts (4pts each)] (a) If x > 0, find the most general antiderivative of f ( x ) = 2 x 3 + 3 x 2 + 5. 2 x 4 4 + 3 x 1 1 + 5 x + C = x 4 2 3 x + 5 x + C (b) Compute integraldisplay / 4 parenleftbigg 2 sin + 1 cos 2 parenrightbigg d integraldisplay / 4 parenleftbigg 2 sin + 1 cos 2 parenrightbigg d = integraldisplay / 4 ( 2 sin + sec 2 ) d = 2 cos + tan | / 4 = 2 cos( / 4) + tan( / 4) ( 2 cos(0) + tan(0)) = ( 2) 2 2 + 1 + 2 = 2 + 3 Math 111 (Fall 2005) Prelim 3 (11/29/2005) 3 (c) If x < 0, evaluate the indefinite integral integraldisplay x 3 + 1 x 4 dx integraldisplay x 3 + 1 x 4 dx = integraldisplay 1 x + 1 x 4 dx = ln | x | + x 3 3 + C = ln | x | 1 3 x 3 + C (d) Compute integraldisplay 4 x e x 2 dx Let u = x 2 . Then du = 2 x dx , and so (1 / 2) du = x dx . Therefore, integraldisplay 4 x e x 2 dx = 1 2 integraldisplay x =4 x =0 e u du = 1 2 integraldisplay 16 e u du = 1 2 [ e u ] 16 = 1 2 ( e 16 e ) = 1 2 ( e 16 1) CONTINUE TO NEXT PAGE Math 111 (Fall 2005) Prelim 3 (11/29/2005) 4 2. [6 pts] Assuming that f ( t ) is differentiable for all t , compute integraldisplay x 2 tf ( t 2 ) dt Let u = t 2 . Then du = 2 t dt . Therefore, integraldisplay x 2 tf ( t 2 ) dt = integraldisplay t = x t =0 f ( u ) du = integraldisplay x 2 f ( u ) du = f ( x 2 ) f (0) (by FTC part 2) 3. [6 pts]...
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math111_2005fall_prelim3sol - Math 111 Prelim 3 November...

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