math111_2003fall_prelim3sol

math111_2003fall_prelim3sol - Math 111 Prelim 3 Nov 20 2003...

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Math 111 Prelim 3 Nov. 20, 2003 Name: SOLUTIONS Instructor: Section: INSTRUCTIONS — READ THIS NOW This test has 9 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 instructor’s name, and your section number right now . Show your work. If you use your calculator to get an answer, say (briefly) what you did. 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 clearly label your work. 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. DERIVATIVE FORMULAS d dx x n = nx n - 1 d dx e x = e x d dx sin( x ) = cos( x ) d dx cos( x ) = - sin( x ) d dx tan( x ) = sec 2 ( x ) d dx csc( x ) = - csc( x ) cot( x ) d dx sec( x ) = sec( x ) tan( x ) d dx cot( x ) = - csc 2 ( x ) INTEGRAL FORMULAS x n dx = x n +1 n +1 + C ( n = - 1) x - 1 dx = ln( | x | ) + C e x dx = e x + C cos( x ) dx = sin( x ) + C sin( x ) dx = - cos( x ) + C sec 2 ( x ) dx = tan( x ) + C OFFICIAL USE ONLY 1. 2. 3. 4. 5. 6. 7. 8. 9. Total: CONTINUE TO THE NEXT PAGE
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Math 111 (Fall 2003) Prelim 3 (11/20/2003) 2 1. [16pts (4pts each)] (a) Find the most general anti-derivative for f ( x ) = 8 x 3 + e x - sin( x ). 2 x 4 + e x + cos x + C (b) Find the most general anti-derivative for g ( x ) = x 2 + 3 x 3 x . x 2 2 + x 3 + C 1 for x < 0 x 2 2 + x 3 + C 2 for x > 0 (c) Compute 2 0 1 - x 2 dx . 2 0 1 - x 2 dx = 1 0 (1 - x 2 ) dx + 2 1 - (1 - x 2 ) dx = x - x 3 3 1 0 + - x + x 3 3 2 1 = 1 - 1 3 - 2 + 8 3 + 1 - 1 3 = 2 (d) Compute e 8 e 2 dx x . e 8 e 2 dx x = ln x | e 8 e 2 = ln e 8 - ln e 2 = 8 - 2 = 6 CONTINUE TO THE NEXT PAGE
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Math 111 (Fall 2003) Prelim 3 (11/20/2003) 3 2. [4pts] Express the following limit as a definite integral on the given interval lim n →∞ n i =1 x * i Δ x, [1 , 4] . (Note: You only need to write the integral, you do not need to evaluate it.) 4 1 x dx 3. [10pts (5pts each)] A Volvo S70 T5 accelerates from a dead stop with its velocity constantly increasing. The table below gives its velocity at several time intervals.
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