246 CHAPTER 4 Polynomial and Rational Functions 37. ( ) = − + H x x 2 1 38. ( ) ( ) = + G x x 2 2 2 39. ( ) = − + + R x x x 1 4 4 2 40. ( ) = − + R x x 1 1 1 41. ( ) ( ) = + − G x x 1 2 3 2 42. ( ) = − + F x x 2 1 1 43. ( ) = − R x x x 4 2 2 44. ( ) = − R x x x 4 In Problems 45–56, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. 45. ( ) = + R x x x 3 4 46. ( ) = + − R x x x 3 5 6 47. ( ) = − − + H x x x x 8 5 6 3 2 48. ( ) = + − − G x x x x 1 5 14 3 2 49. ( ) = − T x x x 1 3 4 50. ( ) = − P x x x 4 1 2 3 51. ( ) = − − − − Q x x x x x 2 5 12 3 11 4 2 2 52. ( ) = + + + + F x x x x x 6 5 2 7 5 2 2 53. ( ) = + − − R x x x x 6 19 7 3 1 2 54. ( ) = + − − R x x x x 8 26 7 4 1 2 55. ( ) = − − G x x x x 1 4 2 56. ( ) = − − F x x x x 16 2 4 2 Applications and Extensions 57. Gravity In physics, it is established that the acceleration due to gravity, g (in m/s2), at a height h meters above sea level is given by ( ) ( ) = × × + g h h 3.99 10 6.374 10 14 6 2 where × 6.374 106 is the radius of Earth in meters. (a) What is the acceleration due to gravity at sea level? (b) The Willis Tower in Chicago, Illinois, is 443 meters tall. What is the acceleration due to gravity at the top of the Willis Tower? (c) The peak of Mount Everest is 8848 meters above sea level.What is the acceleration due to gravity on the peak of Mount Everest? (d) Find the horizontal asymptote of ( ) g h . (e) Solve ( ) = g h 0. How do you interpret your answer? 58. Population Model A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environmentalists declared the insect endangered and transplanted the insect into a protected area.The population P of the insect t months after being transplanted is ( ) ( ) = + + P t t t 50 1 0.5 2 0.01 (a) How many insects were discovered? In other words, what was the population when = t 0? (b) What will the population be after 5 years? (c) Determine the horizontal asymptote of ( ) P t . What is the largest population that the protected area can sustain? 59. Resistance in Parallel Circuits From Ohm’s Law for circuits, it follows that the total resistance Rtot of two components hooked in parallel is given by the equation = + R R R R R tot 1 2 1 2 where R1 and R2 are the individual resistances. (a) Let = R 10 1 ohms, and graph Rtot as a function of R .2 (b) Find and interpret any asymptotes of the graph obtained in part (a). (c) If = R R 2 , 2 1 what value of R1 will yield an Rtot of 17 ohms? 60. Challenge Problem Newton’s Method In calculus you will learn that if ( ) = + + + + − − p x a x a x a x a n n n n 1 1 1 0 is a polynomial function, then the derivative of ( ) p x is ( ) ( ) ′ = + − + + + − − − p x na x n a x a x a 1 2 n n n n 1 1 2 2 1 Newton’s Method is an efficient method for approximating the x-intercepts (or real zeros) of a function, such as ( ) p x . The following steps outline Newton’s Method. Step 1 Select an initial value x0 that is somewhat close to the x-intercept being sought. Step 2 Find values for x using the relation … ( ) ( ) = − ′ = + x x p x p x n 0, 1, 2, n n n n 1 until you get two consecutive values xn and + xn 1 that agree to whatever decimal place accuracy you desire. Step 3 The approximate zero will be + x . n 1 Consider the polynomial ( ) = − − p x x x7 40. 3 (a) Evaluate ( ) p 3 and ( ) p 5 . (b) What might we conclude about a zero of p? (c) Use Newton’s Method to approximate an x-intercept, r, < < r 3 5, of ( ) p x to four decimal places. (d) Use a graphing utility to graph ( ) p x and verify your answer in part (c). (e) Using a graphing utility, evaluate ( ) p r to verify your result. 61. Challenge Problem The standard form of the rational function ( ) = + + ≠ R x mx b cx d c , 0, is ( ) ( ) = − + R x a x h k 1 . To write a rational function in standard form requires polynomial division. (a) Write the rational function ( ) = + − R x x x 2 3 1 in standard form by writing R in the form + Quotient remainder divisor (b) Graph R using transformations. (c) Find the vertical asymptote and the horizontal asymptote of R. 62. Challenge Problem Repeat Problem 61 for the rational function ( ) = − + − R x x x 6 16 2 7 . 63. Challenge Problem Make up a rational function that has = + y x2 1 as an oblique asymptote.
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