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Unformatted text preview: MATH 115 — FINAL EXAM April 25, 2005 NAME: Solution Key INSTRUCTOR: SECTION NO: 1. Do not open this exam until you are told to begin. 2. This exam has 9 pages including this cover. There are 9 questions. 3. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you turn in the exam. 4. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 6. You may use your calculator. You are also allowed two sides of one 3 by 5 note card. 7. If you use graphs or tables to obtain an answer, be certain to provide an explanation and sketch of the graph to make clear how you arrived at your solution. 8. Please turn off all cell phones and other sound devices, and remove all headphones. PROBLEM POINTS SCORE 1 12 2 5 3 5 4 10 5 12 6 12 7 17 8 15 9 12 TOTAL 100 2 1. (3+3+3+3 points) The figure below shows the tangent line approximation of f ( x ) near x = a . x 2 y = 3 x + 9 f ( x ) (a) What are a , f ( a ), and f ′ ( a )? a = 2 f ( a ) = 3 f ′ ( a ) =3 (b) Estimate f (2 . 1). Is this an overestimate or an underestimate? Why? f (2 . 1) ≈ 2.7 is an underestimate because the tangent line approximation of f ( x ) for x > 2 lies below the graph of f ( x ). (c) Estimate f (1 . 98). Is this an overestimate or an underestimate? Why? f (1 . 98) ≈ 3.06 is an overestimate because the tangent line approximation of f ( x ) lies above the graph of f ( x ) for x < 2. (d) Would you expect your estimation for f (2 . 1) or f (1 . 98) to be more accurate? Why? The tangent line approximation is increasingly more accurate the closer one gets to x = 2. Since 2 . 1 2 = 0 . 1 and 2 1 . 98 = 0 . 02, we would expect f (1 . 98) to be more accurate. 3 2. (5 points) Suppose integraldisplay 9 4 (4 f ( x ) + 7) dx = 315. Find integraldisplay 9 4 f ( x ) dx . integraldisplay 9 4 4 f ( x ) dx + integraldisplay 9 4 7 dx = 315 4 integraldisplay 9 4 f ( x ) dx + 35 = 315 4 integraldisplay 9 4 f ( x ) dx = 280 integraldisplay 9 4 f ( x ) dx = 70 3. (5 points) Use the Fundamental Theorem to determine the positive value of b if the area under the graph of f ( x ) = 4 x + 1 between x = 2 and x = b is equal to 11. integraldisplay b 2 (4 x + 1) dx = 11 4 x 2 2  b 2 + x  b 2 = 11 (2 b 2 8) + ( b 2) = 11 2 b 2 + b 21 = 0 (2 b + 7)( b 3) = 0 b = 7 2 , 3 Since b is positive, b = 3. 4 4. (2 points each–no partial credit) Suppose integraldisplay b a f ( x ) dx = 2 and integraldisplay b a g ( x ) dx = 6. Evaluate the following ex pressions, if possible. If the expression cannot be evaluated with what is given, simply indicate ”Insufficient information.” Assume that all functions are continuous on the interval [information....
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This note was uploaded on 02/24/2009 for the course MATH 115 taught by Professor Blakelock during the Fall '08 term at University of Michigan.
 Fall '08
 BLAKELOCK
 Math, Calculus

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