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Final_Solutions

# Final_Solutions - M ATH 115 F INAL E XAM N AME I NSTRUCTOR...

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2 1. (2 points each) Suppose f is a twice-differentiable function. Use the graph of the derivative f , shown below, to answer the following questions. No explanations are required. y = f ( x ) A B C D E F (a) At which of the marked x -values does f attain a global minimum on the interval [A,F]? B (b) At which of the marked x -values does f attain a global maximum on the interval [A,F]? F (c) At which of the marked x -values does f attain a global minimum on the interval [A,F]? A (d) At which of the marked x -values does f attain a global maximum on the interval [A,F]? F (e) At which of the marked x -values does f ′′ attain a global maximum on the interval [A,F]? C (f) For which of the marked x -values does integraldisplay x A f ( t ) dt attain a global minimum on the interval [A,F]? B (g) For which of the marked x -values does integraldisplay x A f ( t ) dt attain a global maximum on the interval [A,F]? F
3 2. (2 points each) Next to each of the functions graphed on the left below, identify which one of the inequalities on the right below best describes the situation. Here, L is the left Riemann sum for integraltext 6 0 f ( x ) dx using three equal subdivisions, and R is the right Riemann sum using three equal subdivisions. [ You may find it helpful to compute L , R , and the integral for each graph. ] 2 4 6 1 2 3 4 - 1 Best described by (f) 2 4 6 1 2 3 4 - 1 Best described by (g) 2 4 6 1 2 3 4 - 1 - 2 Best described by (h) 2 4 6 1 2 3 4 - 1 - 2 Best described by (i) (a) L < R < integraldisplay 6 0 f ( x ) dx (b) L = R < integraldisplay 6 0 f ( x ) dx (c) L < R = integraldisplay 6 0 f ( x ) dx (d) L < integraldisplay 6 0 f ( x ) dx < R (e) L = integraldisplay 6 0 f ( x ) dx < R (f) R < L < integraldisplay 6 0 f ( x ) dx (g) R < L = integraldisplay 6 0 f ( x ) dx (h) R < integraldisplay 6 0 f ( x ) dx < L (i) R = integraldisplay 6 0 f ( x ) dx < L

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4 3. The figure below shows a differentiable function f and the line tangent to the graph at the point (2 , 5) : ( picture not drawn to scale ) (2 , 5) (2 . 5 , 6) f ( x )
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