230
Chapter 3 | Derivatives
velocity at t =2 second.
45. [T] The position in feet of a race car along a straight track after t seconds is modeled by the function s ( t ) =8 t 2 − 1 16 t 3 . a. Find the average velocity of the vehicle over the following time intervals to four decimal places: i. [4, 4.1] ii. [4, 4.01] iii. [4, 4.001] iv. [4, 4.0001] b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at t =4 seconds. 46. [T] The distance in feet that a ball rolls down an incline is modeled by the function s ( t ) =14 t 2 , where t is seconds after the ball begins rolling. a. Find the average velocity of the ball over the following time intervals: i. [5, 5.1] ii. [5, 5.01] b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at t =5 seconds. 47. Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by s = f ( t ) and s = g ( t ), where s is measured in feet and t is measured in seconds. iii. [5, 5.001] iv. [5, 5.0001]
35. s ( t ) = 1 3
t +5
36. s ( t ) = t 2 −2 t 37. s ( t ) =2 t 3 +3
38. s ( t ) = 16 t 2
− 4 t
39. Use the following graph to evaluate a. f ′(1) and b. f ′(6).
40. Use the following graph to evaluate a. f ′(−3) and b. f ′(1.5).
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x = a for each of the given functions.
a. Which vehicle has traveled farther at t =2 seconds? b. What is the approximate velocity of each vehicle at t =3 seconds? c. Which vehicle is traveling faster at t =4 seconds? d. What is true about the positions of the vehicles at t =4 seconds?
41. f ( x ) = x 1/3 , x =0 42. f ( x ) = x 2/3 , x =0
⎧ ⎩ ⎨ 1, x <1 x , x ≥1
43. f ( x ) =
, x =1
44. f ( x ) = | x |
x , x =0
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