Chapter Test 977 (b) Find the average rate of change of revenue from = x 25 to = x 90 wristwatches. (c) Find the average rate of change of revenue from = x 25 to = x 50 wristwatches. (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous rate of change of revenue at = x 25 wristwatches. 40. The function ( ) = + f x x2 3 is nonnegative and continuous on the interval [ ] 0, 4 . (a) Graph f. In (b)–(e), approximate the area A under f from = x 0 to = x 4 as follows: (b) Partition [ ] 0, 4 into four subintervals of equal width and choose u as the left endpoint of each subinterval. (c) Partition [ ] 0, 4 into four subintervals of equal width and choose u as the right endpoint of each subinterval. (d) Partition [ ] 0, 4 into eight subintervals of equal width and choose u as the left endpoint of each subinterval. (e) Partition [ ] 0, 4 into eight subintervals of equal width and choose u as the right endpoint of each subinterval. (f) What is the actual area A? In Problems 41 and 42, each function f is nonnegative and continuous on the given interval. (a) Graph f, indicating the area A under f from a to b. (b) Approximate the area A by partitioning [ ] a b , into three subintervals of equal width and choosing u as the left endpoint of each subinterval. (c) Approximate the area A by partitioning [ ] a b , into six subintervals of equal width and choosing u as the left endpoint of each subinterval. (d) Express the area A as an integral. (e) Use a graphing utility to approximate the integral. 41. ( ) [ ] = − − f x x 4 , 1, 2 2 42. ( ) [ ] = f x x 1 , 1, 4 2 In Problems 43 and 44, an integral is given. (a) What area does the integral represent? (b) Graph the function, and shade the region represented by the integral. (c) Use a graphing utility to approximate this area. 43. x dx 9 1 3 2 ∫ ( ) − − 44. ∫ − e dx x 1 1 The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. In Problems 1–6, find each limit. 1. ( ) − + − → x x lim 3 5 x 3 2 2. − − → + x x lim 2 3 6 x 2 3. − →− x lim 7 3 x 6 4. − − + →− x x x lim 4 5 1 x 1 2 3 5. [ ] ( )( ) − → x x lim 3 2 x 5 2 6. + π → x x lim tan 1 cos x 4 2 Chapter Test 7. Determine the value for k that will make the function continuous at = c 4. ( ) = − + ≤ + > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ f x x x x kx x 9 3 if 4 5 if 4 2 In Problems 8–12, use the graph of ( ) = y f x . y 4 x 4 8. Investigate ( ) → + f x lim x 3 9. Investigate ( ) → − f x lim x 3 10. Investigate ( ) →− f x lim x 2 11. Does the graph suggest that ( ) → f x lim x 1 exists? If so, what is it? If not, explain why not. 12. Determine whether f is continuous at each of the following numbers. If it is not, explain why not. (a) = − x 2 (b) = x 1 (c) = x 3 (d) = x 4 13. Determine where the rational function ( ) = + − − + − R x x x x x x 6 4 24 5 14 3 2 2 is undefined. Determine whether an asymptote or a hole appears at such numbers. 14. For the function ( ) = − − f x x x 4 11 3: 2 (a) Find the derivative of f at = x 2. (b) Find an equation of the tangent line to the graph of f at the point ( ) − 2, 9. (c) Graph f and the tangent line. 15. The function ( ) = − f x x 16 2 is nonnegative and continuous on the interval [ ] 0, 4 . (a) Graph f. (b) Partition [ ] 0, 4 into eight subintervals of equal width and choose u as the left endpoint of each subinterval. Use the partition to approximate the area under the graph of f from = x 0 to = x 4. (c) Find the exact area of the region and compare it to the approximation in part (b).
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