976 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function In Problems 12–15, determine whether f is continuous at c. 12. ( ) = − + = f x x x c 3 2 5 4 2 13. ( ) = − + = − f x x x c 4 2 2 2 14. f x x x x x c 4 2 if 2 4 if 2 2 2 ( ) = − + ≠ − = − ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = − 15. f x x x x x c 4 2 if 2 4 if 2 2 2 ( ) = − + ≠ − − = − ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = − y 2 4 4 6 x 2 In Problems 16–27, use the accompanying graph of ( ) = y f x . 16. What is the domain of f? 17. What is the range of f? 18. Find the x-intercept(s), if any, of f. 19. Find the y-intercept(s), if any, of f. 20. Find ( ) − f 6 and ( ) − f 4 . 21. Find ( ) →− − f x lim . x 4 22. Find ( ) →− + f x lim . x 4 23. Find ( ) → − f x lim . x 2 24. Find ( ) → + f x lim . x 2 25. Does ( ) → f x lim x 0 exist? If it does, what is it? 26. Is f continuous at 0? 27. Is f continuous at 4? 28. Discuss whether ( ) = + − R x x x 4 16 2 is continuous at = − = c c 4 and 4. Use limits to analyze the graph of R at c. 29. Determine where the rational function ( ) = − + − − + R x x x x x x 2 4 8 11 18 3 2 2 is undefined. Determine whether an asymptote or a hole appears at such numbers. In Problems 30–32, find the slope of the tangent line to the graph of f at the given point. Graph f and the tangent line. 30. ( ) ( ) = + f x x x 2 8 at 1, 10 2 31. ( ) ( ) = + − − − f x x x2 3 at 1, 4 2 32. ( ) ( ) = + f x x x at 2, 12 3 2 In Problems 33–35, find the derivative of each function at the number indicated. 33. ( ) = − + f x x4 5 at 3 2 34. ( ) = − f x x x3 at 0 2 35. ( ) = + + f x x x 2 3 2 at 1 2 In Problems 36 and 37, approximate the derivative of each function at the number given using a graphing utility. 36. ( ) = − + − − f x x x x 4 3 6 9at 2 4 3 37. π ( ) = f x x x tan at 6 3 38. Instantaneous Velocity of a Ball In physics it is shown that the height s of a ball thrown straight up with an initial velocity of 96 ft sec from a rooftop 112 feet high is ( ) = = − + + s s t t t 16 96 112 2 where t is the elapsed time that the ball is in the air. The ball misses the rooftop on its way down and eventually strikes the ground. (a) When does the ball strike the ground? That is, how long is the ball in the air? (b) At what time t will the ball pass the rooftop on its way down? (c) What is the average velocity of the ball from = t 0 to = t 2? (d) What is the instantaneous velocity of the ball at time t? (e) What is the instantaneous velocity of the ball at = t 2? (f) When is the instantaneous velocity of the ball equal to zero? (g) What is the instantaneous velocity of the ball as it passes the rooftop on the way down? (h) What is the instantaneous velocity of the ball when it strikes the ground? 39. Instantaneous Rate of Change The following data represent the revenue R (in dollars) received from selling x wristwatches at Wilk’s Watch Shop. Wristwatches, x 0 Revenue, R 0 25 2340 40 3600 50 4375 90 6975 130 8775 160 9600 200 10,000 9900 9375 220 250 12 6 3 9 (a) Find the average rate of change of revenue from = x 25 to = x 130 wristwatches.
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