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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?Find the Maclaurin series for sin(x) using the definition of a Maclaurin series. Check your answer by finding the Maclaurin series for sin(x) using Table 11.5.Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 9. f(x)=1x2,a=1Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 10. f(x)=1x2,a=1Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 11. f(x) = ex, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 12. f(x) = cos 2x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 13. f(x)=2(1x)3,a=0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 14. f(x) = x sin x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 15. f(x) = (1 + x2)1, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 16. f(x) = ln (1 + 4x), a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 17. f(x) = e2x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 18. f(x) = (1 + 2x)1, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 19. f(x)=tan1x2,a=0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 20. f(x) = sin 3x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 21. f(x) = 3x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 22. f(x) = log3(x + 1), a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 23. f(x) = cosh 3x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 24. f(x) = sinh 2x, a = 0Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 25. f(x) = ln (x 2), a = 3Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. 26. f(x) = ex, a = 2Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 21. f(x) = sin x, a = /2Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 22. f(x) = cos x, a =Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 23. f(x) = 1/x, a = 1Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 24. f(x) = 1/x, a = 2Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 25. f(x) = ln x, a = 3Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 26. f(x) = ex, a = ln 2Taylor series centered at a 0 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 27. f(x) = 2x, a = 1Taylor series a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. 34. f(x) = x ln x x + 1; a = 1Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 35. ln (1 + x2)Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 36. sin x2Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 37. 112xManipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 38. ln (1 + 2x)Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 39. {ex1xifx11ifx=1Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 40. cos x3Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 41. (1 + x4)1Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 42. x tan1 x2Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 43. sinh x244EBinomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. 39. f(x) = (1 + x)2; approximate 1/1.21 = 1/1.12.Binomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. 40. f(x)=1+x; approximate 1.06.Binomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. 47. f(x)=41+x; approximate41.12.Binomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. 42. f(x) = (1 + x)3; approximate 1/1.331 = 1/1.13.Binomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. 43. f(x) = (1 + x)2/3; approximate 1.182/3.Binomial series a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. 44. f(x) = (1 + x)2/3; approximate 1.022/3.Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series 1+x=1+x2x28+x316...,for-1x1 51. 1+x2Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence forthe new series (Theorem 11.4 is useful). Use the Maclaurin series 1+x=1+x2x28+x316, for 1x1. 52. 4+xWorking with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence forthe new series (Theorem 11.4 is useful). Use the Maclaurin series 1+x=1+x2x28+x316, for 1x1. 53. 99x54EWorking with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence forthe new series (Theorem 11.4 is useful). Use the Maclaurin series 1+x=1+x2x28+x316, for 1x1. 55. a2+x2, a 051-56 Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the Allowing functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series
for −1 ≤ x ≤ 1.
56.
Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 51. (1 + 4x)2Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 52. 1(14x)2Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 53. 1(4+x2)2Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 54. (x2 4x + 5)2Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 55. 1(3+4x)2Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series (1+x)2=12x+3x24x3+,for1x1. 56. 1(1+4x2)2Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that limnRn(x)=0 for all x in the interval of convergence. 57. f(x) = sin x, a = 064ERemainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that limnRn(x)=0 for all x in the interval of convergence. 59. f(x) = ex, a = 0Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that limnRn(x)=0 for all x in the interval of convergence. 60. f(x) = cos x, a = /2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function f(x)=x has a Taylor series centered at 0. b. The function f(x) = csc x has a Taylor series centered at /2. c. If f has a Taylor series that converges only on (2, 2), then f(x2) has a Taylor series that also converges only on (2, 2). d. If p(x) is the Taylor series for f centered at 0, then p(x 1) is the Taylor series for f centered at 1. e. The Taylor series for an even function about 0 has only even powers of x.Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 62. f(x) = cos 2x + 2 sin xAny method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 63. f(x)=ex+ex2Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 64. f(x)={sinxxifx01ifx=0Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 65. f(x) = (l + x2)2/3Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 66. f(x) = x2 cos x2Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 67. f(x)=1x2Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 68. f(x) = bx, for b 0, b 1Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. 69. f(x)=1x4+2x2+1Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. 70. f(x)=x with a = 36; approximate 39.Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. 72. f(x)=1/x with a = 4; approximate 1/3.80EInteger coefficients Show that the first five nonzero coefficients of the Taylor series (binomial series) for f(x)=1+4x about 0 are integers. (In fact, all the coefficients are integers.)Choosing a good center Suppose you want to approximate 72 using four terms of a Taylor series. Compare the accuracy of the approximations obtained using Taylor series for x centered at 64 and 81.Alternative means By comparing the first four terms, show that the Maclaurin series for sin2 x can be found (a) by squaring the Maclaurin series for sin x, (b) by using the identity sin2 x = (1 cos 2x)/2, or (c) by computing the coefficients using the definition.Alternative means By comparing the first four terms, show that the Maclaurin series for cos2 x can be found (a) by squaring the Maclaurin series for cos x, (b) by using the identity cos2 x = (1 + cos 2x)/2, or (c) by computing the coefficients using the definition.85EComposition of series Use composition of series to find the first three terms of the Maclaurin series for the following functions. a. esin x b. etan x c. 1+sin2x87EApproximations Choose a Taylor series and center point to approximate the following quantities with an error of 104 or less. 84. sin (0.98)Different approximation strategies Suppose you want to approximate 1283 to within 10-4 of the exact value. a. Use a Taylor polynomial for f(x) = (125 + x)1/3 centered at 0. b. Use a Taylor polynomial for f(x) = x1/3 centered at 0. c. Compare the two approaches Are they equivalent?90E91EUse the Taylor series sin x = x - x3/6+ to verify that limx0(sinx)/x=1.2QC3QCExplain the strategy presented in this section for evaluating a limit of the form limxaf(x)/g(x), where f and g have Taylor series centered at a.Explain the method presented in this section for approximating abf(x)dx, where f has a Taylor series with an interval of convergence centered at a that includes b.How would you approximate e0.6 using the Taylor series for ex?Use the Taylor series for cos x centered at 0 to verify that limx01cosxx=0.Use the Taylor series for sinh X and cosh X to verify that ddxsinhx=coshx.What condition must be met by a function f for it to have a Taylor series centered at a?Limits Evaluate the following limits using Taylor series. 7. limx0ex1xLimits Evaluate the following limits using Taylor series. 8. limx0tan1xxx3Limits Evaluate the following limits using Taylor series. 9. limx0xln(1x)x2Limits Evaluate the following limits using Taylor series. 10. limx0sin2xxLimits Evaluate the following limits using Taylor series. 11. limx0exexxLimits Evaluate the following limits using Taylor series. 12. limx01+xex4x2Limits Evaluate the following limits using Taylor series. 13. limx02cos2x2+4x22x4Limits Evaluate the following limits using Taylor series. 14. limxxsin1xLimits Evaluate the following limits using Taylor series. 15. limx0ln(1+x)x+x2/2x3Limits Evaluate the following limits using Taylor series. 16. limx4x216ln(x3)Limits Evaluate the following limits using Taylor series. 17. limx03tan1x3x+x3x5Limits Evaluate the following limits using Taylor series. 18. limx01+x1(x/2)4x2Limits Evaluate the following limits using Taylor series. 19. limx012x8x36sin2xx5Limits Evaluate the following limits using Taylor series. 20. limx1x1lnxLimits Evaluate the following limits using Taylor series. 21. limx2x2ln(x1)Limits Evaluate the following limits using Taylor series. 22. limxx(e1/x1)Limits Evaluate the following limits using Taylor series. 23. limx0e2x4ex/2+32x2Limits Evaluate the following limits using Taylor series. 24. limx0(12x)1/2ex8x2Power series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 25. f(x) = exPower series for derivatives a. Differentiate the Taylor series centered at 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 26. f(x) = cos xPower series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 27. f(x) = ln (1 + x)Power series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 28. f(x) = sin x2Power series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 29. f(x) = e2xPower series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 30. f(x) = (1 x)1Power series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 31. f(x) = tan1 xPower series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. 32. f(x) = ln (1 x)Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. 33. y(t) y = 0, y(0) = 2Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. 34. y(t) + 4y = 8, y(0) = 0Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. 35. y(t) 3y = 10, y(0) = 2Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. 36. y(t) = 6y + 9, y(0) = 2Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 37. 00.25ex2dxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 38. 00.2sinx2dxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 39. 0.350.35cos2x2dxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 40.00.21+x4dxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 41. 00.35tan1xdxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 42. 00.4ln(1+x2)dxApproximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 43. 00.5dx1+x6Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 104. 44. 00.2ln(1+t)tdtApproximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 45. e2Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 46. eApproximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 47. cos 2Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 48. sin 1Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 49. ln32Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. 50. tan112Evaluating an infinite series Let f(x) = (ex 1)/x, for x 0, and f(0) = 1. Use the Taylor series for f about 0 and evaluate f(1) to find the value of k=01(k+1)!.52EEvaluating an infinite series Write the Taylor series for f(x) = ln (1 + x) about 0 and find its interval of convergence. Assume the Taylor series converges to f on the interval of convergence. Evaluate f(1) to find the value of k=1(1)k+1k (the alternating harmonic series).54ERepresenting functions by power series Identify the functions represented by the following power series. 55. k=0xk2kRepresenting functions by power series Identify the functions represented by the following power series. 56. k=0(1)kxk3kRepresenting functions by power series Identify the functions represented by the following power series. 57. k=0(1)kx2k4kRepresenting functions by power series Identify the functions represented by the following power series. 58. k=02kx2k+1Representing functions by power series Identify the functions represented by the following power series. 59. k=1xkkRepresenting functions by power series Identify the functions represented by the following power series. 60. k=0(1)kxk+14kRepresenting functions by power series Identify the functions represented by the following power series. 61. k=1(1)kkxk+13kRepresenting functions by power series Identify the functions represented by the following power series. 62. k=1x2kkRepresenting functions by power series Identify the functions represented by the following power series. 63. k=2k(k1)xk3kRepresenting functions by power series Identify the functions represented by the following power series. 64. k=2xkk(k1)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate 02dx1x, one could expand the integrand in a Taylor series and integrate term by term. b. To approximate /3, one could substitute x=3 into the Taylor series for tan1 x. c. k=0(ln2)kk!=2.Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). 66. limx0eax1xLimits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). 67. limx0sinaxsinbxLimits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). 68. limx0sinaxtan1axbx3A limit by Taylor series Use Taylor series to evaluate limx0(sinxx)1/x2.70E71E72E73E74E75EProbability: sudden-death playoff Teams A and B go into sudden-death overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a1/6 chance of scoring when it has the ball, and Team A has the ball first. a. The probability that Team A ultimately wins is k=016(56)2kEvaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is 16k=0k(56)k1Evaluate this series.Elliptic integrals The period of an undamped pendulum is given by T=lg40/2d1k2sin2=lg4F(k), where l is the length of the pendulum, g = 9.8 m/s2 is the acceleration due to gravity, k = sin 02, and 0 is the initial angular displacement of the pendulum (in radians). The integral in this formula F(k) iscalled an elliptic integral, and it cannot be evaluated analytically. Approximate F(0.1) by expanding theintegrand in a Taylor (binomial) series and integrating term by term.Sine integral function The function Si(x)=0xf(t)dt, where f(t)={sinttift01ift=0, is called the sine integral function. a. Expand the integrand in a Taylor series centered at 0. b. Integrate the series to find a Taylor series for Si. c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximationdoes not exceed 10-3.Fresnel integrals The theory of optics gives rise to the two Fresnel integrals S(x)=0xsint2dtandC(x)=0xcost2dt. a. Compute S(x) and C(x). b. Expand sin t2 and cos t2 in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for S and C. c. Use the polynomials in part (b) to approximate S(0.05) and C(0.25). d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 104? e. How many terms of the Maclaurin series are required to approximate C(0.25) with an error no greater than 106?Error function An essential function in statistics and the study of the normal distribution is the error function erf(x)=20xet2dt. a. Compute the derivative of erf (x). b. Expand et2 in a Maclaurin series; then integrate to find the first four nonzero terms of the Maclaurin series for erf. c. Use the polynomial in part (b) to approximate erf (0.15) and erf (0.09). d. Estimate the error in the approximations of part (c).81E83E84EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Let pn be the nth-order Taylor polynomial for f centered at 2. The approximation p3(2.1) f(2.1)is likely to be more accurate than the approximation p2(2.2) f(2.2). b. If the Taylor series for f centered at 3 has a radius of convergence of 6, then the interval ofconvergence is [3, 9]. c. The interval of convergence of the power series ckxkcould be (7/3, 7/3). d. The Maclaurin series for f(x) = (1 + x)12 has a finite number of nonzero terms. e. If the power series ck(x3)khas a radius of convergence of R = 4 and converges at theendpoints of its interval of convergence, then its interval of convergence is [1, 7].2RETaylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. 3. f(x) = cos3 x, n = 2, a = 0Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. 4. f(x) = cos1 x, n = 2, a = 12Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = e1/x-1, n =2, a = 1Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = esin x , n = 2, a = 0Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = cos (ln x), n = 2 , a = 1Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = sinh (3x), n = 3, a = 09REApproximations a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point b. Use the Taylor polynomials to approximate the given expression. Make a table showing theapproximations and the absolute error in these approximations using a calculator for the exact functionvalue. 10. f(x) = cos x, a = 0; cos (0.08)Approximations a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point b. Use the Taylor polynomials to approximate the given expression. Make a table showing theapproximations and the absolute error in these approximations using a calculator for the exact functionvalue. f(x) = ex, a = 0; e0.08Approximations a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point b. Use the Taylor polynomials to approximate the given expression. Make a table showing theapproximations and the absolute error in these approximations using a calculator for the exact functionvalue. f(x) = 1+x, a = 0; 1.0813REEstimating remainders Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.) 14. f(x) = ex; bound R3(x), for |x| 1.Estimating remainders Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.) 15. f(x) = sin x; bound R3(x), for |x| .Estimating remainders Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.) 16. f(x) = ln (1 x); bound R3(x), for |x| 1/2.17RE18RERadius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, whenappropriate. k=1(1)(x+1)2kk!Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, whenappropriate. 20. k=1(x1)kk5k21RE22RERadius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, whenappropriate. 23. k=1(x+2)2kInkk24RE25RE26RE27RE28REPower series from the geometric series Use the geometric series k=0xk=11x, for |x| 1, to determine the Maclaurin series and the interval of convergence for the following functions. 25. f(x)=11x2Power series from the geometric series Use the geometric series k=0xk=11x, for |x| 1, to determine the Maclaurin series and the interval of convergence for the following functions. 26. f(x)=11+x3Power series from the geometric series Use the geometric series k=0xk=11x, for |x| 1, to determine the Maclaurin series and the interval of convergence for the following functions. 27. f(x)=11+5x32RE33REPower series from the geometric series Use the geometric series k=0xk=11x, for |x| 1, to determine the Maclaurin series and the interval of convergence for the following functions. 30. f(x) = ln (1 4x)Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. 31. f(x) = e3x, a = 036RETaylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. 33. f(x) = cos x, a = /2Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. 34. f(x)=x21+x,a=0Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. 35. f(x) = tan1 4x, a = 0Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. 36. f(x) = sin 2x, a = /241RE42RE43RE44REBinomial series Write out the first three terms of the Maclaurin series for the following functions. 41. f(x) = (l + x/2)346RE47REConvergence Write the remainder term Rn(x) for the Taylor series for the following functions centered at the given point a. Then show that limnRn(x)=0, for all x in the given interval. 45. f(x)=ln(1+x),a=0,12x12Limits by power series Use Taylor series to evaluate the following limits. 47. limx0x2/21+cosxx4Limits by power series Use Taylor series to evaluate the following limits. 48. limx02sinxtan1xx2x5Limits by power series Use Taylor series to evaluate the following limits. 49. limx4ln(x3)x216Limits by power series Use Taylor series to evaluate the following limits. 50. limx01+2x1xx2Limits by power series Use Taylor series to evaluate the following limits. 51. limx0secxcosxx2x4(Hint:TheMaclaurinseriesforsecxis1+x22+5x424+61x6720+.)Limits by power series Use Taylor series to evaluate the following limits. 54. limx0(1+x)2316x2x2Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 103. 53. 01/2ex2dx56REDefinite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 103. 55. 01xcosxdx58REApproximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series. 57. 11960REApproximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series. 59. tan1(13)62RE63RERejected quarters The probability that a random quarter is not rejected by a vending machine is given by the integral 11.4 00.14e102x2dx (assuming the weights of quarters are normally distributed with a mean of 5.670 g and a standard deviation of 0.07 g). Estimate the value of the integral by using the first two terms of the Maclaurin series fore102x2.65REGraphing Taylor polynomials Consider the function f(x)=(1+x)4. a. Find the Taylor polynomials p0, p1, p2, and p3 centered at 0. b. Use a graphing utility to plot the Taylor polynomials and f, for 1 x 1. c. For each Taylor polynomial, give the interval on which its graph appears indistinguishable from the graph of f.Identify the graph generated by the parametric equations x = t2, y = t, for 10 t 10.2QCDescribe the curve generated by x = 3 + 2t, y = 12 6t, for t .Find parametric equations for the line segment that goes from Q(0, 3) to P(2, 0).Use Theorem 12.1 to find the slope of the line x = 4t, y = 2t, for t .Use the arc length formula to find the length of the line x = t, y = t, for 0 t 1.Explain how a pair of parametric equations generates a curve in the xy-plane.2E3EGive parametric equations that generate the line with slope 2 passing through (1, 3).Find parametric equations for the complete parabola x = y2. Answers are not unique.Describe the similarities between the graphs of the parametric equations x = sin2 t, y = sin t, for 0 t /2, and x = sin2 t, y = sin t, for /2 t . Begin by eliminating the parameter t to obtain an equation in x and y.Find the slope of the parametric curve x = 2t3 + 1, y = 3t2, for t , at the point corresponding to t = 2.8EFind three different pairs of parametric equations for the line segment that starts at (0, 0) and ends at (6, 6).Use calculus to find the arc length of the line segment x = 3t + 1, y = 4t, for 0 t 1. Check your work by finding the distance between the endpoints of the line segment.11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27EWorking with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. 28. x = 1 3 sin 4t, y = 2 + 3 cos 4t; 0 t 1/229E30EEliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y. 83. x = 2 sin 8t, y = 2 cos 8tEliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y. 84. x = sin 8t, y = 2 cos 8t33E34E35E36EParametric equations of circles Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique. 27. A circle centered at the origin with radius 4, generated counterclockwiseParametric equations of circles Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique. 28. A circle centered at the origin with radius 12, generated clockwise with initial point (0, 12)Parametric equations of circles Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique. 29. A circle centered at (2, 3) with radius 1, generated counterclockwiseParametric equations of circles Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique. 31. A circle centered at (2, 3) with radius 8, generated clockwise41ECurves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. 42. The line segment starting at P(1, 3) and ending at Q(6, 16)Curves to parametric equations Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. 45. The segment of the parabola y = 2x2 4, where 1 x 5Curves to parametric equations Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. 46. The complete curve x = y3 3y45ECurves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. 46. The horizontal line segment starting at P(8, 2) and ending at Q(2, 2)47E48E49ECurves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. 50. The left half of the parabola y x2 + 1, originating at (0, 1)Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. 51. The upper half of the parabola x = y2, originating at (0, 0)Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. 52. The lower half of the circle centered at (2, 2) with radius 6, oriented in the counterclockwise directionCircular motion Find parametric equations that describe the circular path of the following objects. For Exercises 5355, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle. 53. A go-cart moves counterclockwise with constant speed around a circular track of radius 400 m, completing a lap in 1.5 min.Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 5355, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle. 54. The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 5355, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle. 55. A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 5355, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle. 56. A Ferris wheel has a radius of 20 m and completes a revolution in the clockwise direction at constant speed in 3 min. Assume x and y measure the horizontal and vertical positions of a seat on the Ferris wheel relative to a coordinate system whose origin is at the low point of the wheel. Assume the seat begins moving at the origin.More parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 49. Spiral x = t cos t, y = t sin t; t 0More parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 50. Witch of Agnesi x = 2 cot t, y = 1 cos 2tMore parametric curves Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 51. Folium of Descartes x=3t1+t3,y=3t21+t3