W05FinalExam_Solutions

# W05FinalExam_Solutions - MATH 115 FINAL EXAM NAME...

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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.

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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 [ a, b ].
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