450 CHAPTER 6 Trigonometric Functions Mixed Practice In Problems 41–44, find the average rate of change of f from 0 to π 6 . 41. ( ) = f x x tan 42. ( ) = f x x sec 43. ( ) ( ) = f x x tan 2 44. ( ) ( ) = f x x sec 2 Mixed Practice In Problems 45–48, find ( )( ) f g x and ( )( ) g f x , and graph each of these functions. Mixed Practice In Problems 49 and 50, graph each function. 49. π π π π ( ) = ≤ < = < ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ f x x x x x x tan if 0 2 0 if 2 sec if 2 50. π π π π ( ) = < < = < < ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ g x x x x x x csc if 0 0 if cot if 2 Applications and Extensions 51. Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the figure. u 4 ft 3 ft L (a) Show that the length L of the ladder shown as a function of the angle θ is θ θ θ ( ) = + L 3sec 4csc (b) Graph θ θ π ( ) = < < L L , 0 2 . (c) For what value of θ is L the least? (d) What is the length of the longest ladder that can be carried around the corner? Why is this also the least value of L? 52. A Rotating Beacon Suppose that a fire truck is parked in front of a building as shown in the figure. FIRE LA d 10 ft A The beacon light on top of the fire truck is located 10 feet from the wall and has a light on each side. If the beacon light rotates 1 revolution every 2 seconds, then a model for determining the distance d, in feet, that the beacon of light is from point A on the wall after t seconds is given by π ( ) ( ) = d t t 10 tan (a) Graph π ( ) ( ) = d t t 10tan for ≤ ≤ t 0 2. (b) For what values of t is the function undefined? Explain what this means in terms of the beam of light on the wall. (c) Fill in the following table. t 0 0.1 0.2 0.3 0.4 π ( ) ( ) = d t t 10tan (d) Compute ( ) ( ) ( ) ( ) − − − − d d d d 0.1 0 0.1 0 , 0.2 0.1 0.2 0.1 , and so on, for each consecutive value of t. These are called first differences. (e) Interpret the first differences found in part (d). What is happening to the speed of the beam of light as d increases? 53. Exploration Graph π ( ) = = − + y x y x tan and cot 2 Do you think that π ( ) = − + x x tan cot 2 ? 54. Challenge Problem What are the domain and the range of ( ) ( ) = f x x log tan ? Find any vertical asymptotes. 55. Challenge Problem What are the domain and the range of ( ) = f x x ln sin ? Find any vertical asymptotes. 45. ( ) ( ) = = f x x g x x tan 4 46. ( ) ( ) = = f x x g x x 2 sec 1 2 47. ( ) ( ) = − = f x x g x x 2 cot 48. ( ) ( ) = = f x x g x x 1 2 2csc
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