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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Applying convergence tests Determine whether the following series converge. Justify your answers. 83. j=21jln10jApplying convergence tests Determine whether the following series converge. Justify your answers. 84. k=1tan11k3Applying convergence tests Determine whether the following series converge. Justify your answers. 85. 123+145+167+189+...Applying convergence tests Determine whether the following series converge. Justify your answers. 86. k=1akk!,a087EA few more series Determine whether the following series converge. Justify your answers. 88. k=1(k4+1k2)A few more series Determine whether the following series converge. Justify your answers. 89. k=1k1+2+...+kA few more series Determine whether the following series converge. Justify your answers. 90. k=1(13+23+33+...+k3k5)A few more series Determine whether the following series converge. Justify your answers. 91. k=0ln(2k+12k+4)A few more series Determine whether the following series converge. Justify your answers. 92. k=1110lnkA few more series Determine whether the following series converge. Justify your answers. 93. k=112lnk+2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The terms of the sequence {an} increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series 1/k approach zero, so the series converges. c. The terms of the sequence of partial sums of the series approach so the infinite series ak approach 52, so the infinite series converges to 52. d. An alternating series that converges absolutely must converge conditionally. e. The sequence an=n2n2+1 converges. f. The sequence an=(1)nn2n2+1 converges. g. The series k=1k2k2+1 converges. h. The sequence of partial sums associated with the series k=1k2k2+1 converges.Related sequences a. Find the limit of the sequence {12n3+4n6n3+5}. b. Explain why the sequence {(1)n12n3+4n6n3+5} diverges.Geometric sums Evaluate the geometric sums k=09(0.2)k and k=29(0.2)k.A savings plan Suppose you open a savings account by depositing 100. The account earns interest at an annual rate of 3% per year (0.25% per month). At the end of each month, you earn interest on the current balance, and then you deposit 100. Let Bn be the balance at the beginning of the nth month, where B0 = 100. a. Find a recurrence relation for the sequence {Bn}. b. Find an explicit formula that gives Bn, for n = 0, 1, 2, 3, .... c. Find B30, the balance at the beginning of the 30th month.Partial sums Let Sn be the nth partial sum of k=1ak=8. Find limxak and limnSn.Find the value of r for which k=03rk=6.Sequences versus series 53. Give an example (if possible) of a sequence {ak} that converges, while the series k=1ak diverges.Sequences versus series 54. Give an example (if possible) of a series k=1ak that converges, while the sequence {ak} diverges.Sequences versus series a. Find the limit of the sequence {(45)k} b. Evaluate k=0(45)k.Sequences versus series a. Find the limit of the sequence {1k1k+1}. b. Evaluate k=1(1k1k+1).Sequences versus series 55. a. Does the sequence {kk+1} converge? Why or why not? b. Does the series k=1kk+1 converge? Why or why not?12RELimits of sequences Evaluate the limit of the sequence or state that it does not exist. 13. an = (1)n3n3+4n6n3+5Limits of sequences Evaluate the limit of the sequence or state that it does not exist. 14. an=(2n+2)!(2n)!n215RELimits of sequences Evaluate the limit of the sequence or state that it does not exist. 2. an=n2+44n4+1Limits of sequences Evaluate the limit of the sequence or state that it does not exist. 3. an=8nn!Limits of sequences Evaluate the limit of the sequence or state that it does not exist. 4. an=(1+3n)2n19RELimits of sequences Evaluate the limit of the sequence or state that it does not exist. 6. an=nn21Limits of sequences Evaluate the limit of the sequence or state that it does not exist. 7. an=(1n)1/lnnLimits of sequences Evaluate the limit of the sequence or state that it does not exist. 8. an=sinn6Limits of sequences Evaluate the limit of the sequence or state that it does not exist. 9. an=(1)n0.9nLimits of sequences Evaluate the limit of the sequence or state that it does not exist. 24. an = 2tan1nRecursively defined sequences The following sequences {an}n=0 are defined by a recurrence relation. Assume each sequence is monotonic and bounded. a. Find the first five terms a0, a1, , a4 of each sequence. b. Determine the limit of each sequence. 25. an+1=12an + 8; a0 = 80Recursively defined sequences The following sequences {an}n=0 are defined by a recurrence relation. Assume each sequence is monotonic and bounded. a. Find the first five terms a0, a1, , a4 of each sequence. b. Determine the limit of each sequence. 26. an+1=13an+34; a0 = 81Evaluating series Evaluate the following infinite series or state that the series diverges. 13. k=13(1.001)kEvaluating series Evaluate the following infinite series or state that the series diverges. 12. k=1(910)kEvaluating series Evaluate the following infinite series or state that the series diverges. 29. k=0((13)k+(43)k)30RE31REEvaluating series Evaluate the following infinite series or state that the series diverges. 14. k=0(15)k33RE34REEvaluating series Evaluate the following infinite series or state that the series diverges. 35. k=19(3k2)(3k+1)36REEvaluating series Evaluate the following infinite series or state that the series diverges. 19. k=12k3k+2Express 0.29292929... as a ratio of two integers.Express 0.314141414... as a ratio of two integers.Finding steady states through sequences Suppose you take 100 mg of aspirin once per day. Assume the aspirin has a half-life of one day; that is, every day, half of the aspirin in your blood is eliminated. Assume dn is the amount of aspirin in your blood after the nth dose, where d1 = 100. a. Find a recurrence relation for the sequence {dn}. b. Assuming the sequence {dn} converges, find the long-term (steady-state) amount of aspirin in your blood.Finding steady states through sequences Suppose you take 100 mg of aspirin once per day. Assume the aspirin has a half-life of one day; that is, every day, half of the aspirin in your blood is eliminated. Assume dn is the amount of aspirin in your blood after the nth dose, where d1 = 100. a. Find a recurrence relation for the sequence {dn}. b. Assuming the sequence {dn} converges, find the long-term (steady-state) amount of aspirin in your blood. 41. Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. 42. k=1kk5/2+1Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 23. k=1k2/344REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. k=12k4+k3+13k4+4Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. k=1kk/22kk!47RE48REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. k=1k49k12+2Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. 50. k=1k!kkConvergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 25. k=12kekConvergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 26. k=1(kk+3)2kConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 53. k=1(k2k+3)kConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 54. k=1sin1k4Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. 55. k=1k!ekkk56RE57RE58REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 59. j=024j(2j+1)!60REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 61. k=3lnkk3/2Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. 62. k=1k2(2k1)!Convergence or divergence Use a convergence test of your choice to determine whether the following series converge. 63. k=132+ek64REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 31. k=1kkk3Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 32. k=111+lnkConvergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge. 33. k=1k5ek68REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 69. k=1(1cos1k)70REConvergence or divergence Use a convergence test of your choice to determine whether the following series converge. 71. k=1(1cos1k)272RE73RE74RE75RE76REAbsolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 77. k=1(1)k+1k3/778REAbsolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 79. k=2(1)kk2180RE81REAbsolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 82. k=1(1)kk21Absolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 83. k=1(1)kkekAbsolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 84. k=0(1)kek+ek85REAbsolute or conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 86. k=2(1)kklnk87RE88RE89RELower and upper bounds of a series For each convergent series and given value of n. complete the following. a. Use the nth partial sum Sn to estimate the value of the series. b. Find an upper bound for the remainder Rn. c. Find lower and upper bounds (Ln and Un, respectively) for the exact value of the series. 90. k=11k2+1; n = 20Estimate the value of the series k=11(2k+5)3 to within 104 of its exact value.92REError in a finite alternating sum How many terms of the series k=1(1)k+1k4 must be summed to ensure that the approximation error is less than 108?Equations involving series Solve the following equations for x. a. k=0ekx=2 b. k=0(3x)k=495RE96REPages of circles On page 1 of a book, there is one circle of radius 1. On page 2, there are two circles of radius 12. On page n, there are 2n1 circles of radius 2n+1. a. What is the sum of the areas of the circles on page n of the book? b. Assuming the book continues indefinitely (n ), what is the sum of the areas of all the circles in the book?98RE99RE100REBouncing ball for time Suppose a rubber ball, when dropped from a given height, returns to a fraction p of that height. In the absence of air resistance, a ball dropped from a height h requires 2hg seconds to fall to the ground, where g 9.8 m/s2 is the acceleration due to gravity. The time taken to bounce up to a given height equals the time to fall from that height to the ground. How long does it take for a ball dropped from 10 m to come to rest?Verify that p3 satisfies p3(k)(a)=f(k)(a), for k = 0, 1, 2, and 3.Verify the following properties for f(x) = sin x and p3(x) = x x3/6: f(0) = p3(0), f(0) = p3(0), f(0) = p3(0), and f(0) = p3(0).Why do the Taylor polynomials for sin x centered at 0 consist only of odd powers of x?Write out the next two Taylor polynomials p4 and p5 for f(x) = ex in Example 3. Example 3 Taylor Polynomials for ex a. Find the Taylor polynomials of order n = 0, 1, 2, and 3 for f(x) = ex centered at 0. Graph f and the polynomials. b. Use the polynomials in part (a) to approximate e0.1 and e0.25. Find the absolute errors, |f(x) pn(x)|, in the approximations. Use calculator values for the exact values of f.At what point would you center the Taylor polynomials for x and x4 to approximate 51 and 154, respectively?In Example 7, find an approximate upper bound for R7(0.45). Example 7 Estimating the Remainder for ex Find a bound on the remainder in approximating e0.45 using the Taylor polynomial of order n = 6 for f(x) = ex centered at 0.Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.The first three Taylor polynomials for f(x)=1+x centered at 0 are p0(x) = 1, p1(x)=1+x2, and p2(x)=1+x2x28. Find three approximations to 1.1.Suppose f(0) = 1, f(0) = 2, and f(0) = 1. Find the quadrate approximate polynomial for f centered at 0 and use it to approximate f(0, 1).Suppose f(0) = 1, f(0) = 0, f"(0) = 2, and f(3)(0) = 6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0, 2).How is the remainder Rn(x) in a Taylor polynomial defined?Suppose f(2) = 1, f(2) = 1, f(2) = 0, and f3(2) = 12. Find the third-order Taylor polynomial for f centered at 2 and use this polynomial to estimate f(1.9).Suppose you want to estimate 26 using a fourth-order Taylor polynomial centered at x = a for f(x)=x. Choose an appropriate value for the center a.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 7. f(x) = 8x3/2, a = 1; approximate 8 1.13/2.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 8. f(x)=1x, a = 1; approximate 11.05.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 11. f(x) = e2x, a = 0; approximate e0.2.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 10. f(x)=x, a = 4; approximate 3.9.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 11. f(x) = (1 + x)1, a = 0; approximate 11.05.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 12. f(x) = cos x, a = /4; approximate cos (0.24).Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 13. f(x) = x1/3, a = 8; approximate 7.51/3.Linear and quadratic approximation a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 14. f(x) = tan1 x, a = 0: approximate tan1 0.1.Find the Taylor polynomials p1, , p4 centered at a = 0 for f(x) = cos 6x.Find the Taylor polynomials p1, , p5 centered at a = 0 for f(x) = ex.Find the Taylor polynomials p3, , p4 centered at a = 0 for f(x) = (1 + x)3.Find the Taylor polynomials p4 and p5 centered at a = /6 for f(x) = cos x.Find the Taylor polynomials p1, p2, and p3 centered at a = 1 for f(x) = x3.Find the Taylor polynomials p3 and p4 centered at a = 1 for f(x)=8x.Find the Taylor polynomial p3 centered at a = e for f(x) = ln x.Find the Taylor polynomial p2 centered at a = 8 for f(x)=x3.Graphing Taylor polynomials a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 1 and n = 2. b. Graph the Taylor polynomials and the function. 25. f(x) = ln (1 x), a = 0Graphing Taylor polynomials a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 1 and n = 2. b. Graph the Taylor polynomials and the function. 26. f(x) = ln (1 + x)1/2, a = 0Graphing Taylor polynomials a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 1 and n = 2. b. Graph the Taylor polynomials and the function. 27. f(x) = sin x, a=4Graphing Taylor polynomials a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 1 and n = 2. b. Graph the Taylor polynomials and the function. 28. f(x)=x, a = 9Approximations with Taylor polynomials a. Use the given Taylor polynomial p2 to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 23. Approximate 1.05 using f(x)=1+x and p2(x) = 1 + x/2 x2/8.30EApproximations with Taylor polynomials a. Use the given Taylor polynomial p2 to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 27. Approximate e0.15 using f(x) = ex and p2(x) = 1 x + x2/2.Approximations with Taylor polynomials a. Use the given Taylor polynomial p2 to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 26. Approximate ln 1.06 using f(x) = ln (1 + x) and p2(x) = x x3/2.Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 39. e0.12Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 40. cos (0.2)Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 41. tan (0.1)Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 42. ln 1.05Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 43. 1.06Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 44. 794Approximations with Taylor polynomials a. Approximate the given quantities using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. 47. sinh 0.540ERemainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 49. f(x) = sin x, a = 0Remainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 50. f(x) = cos 2x, a = 0Remainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 43. f(x) = ex, a = 0Remainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 52. f(x) = cos x, a = /2Remainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 53. f(x) = sin x, a = /2Remainders Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n. 54. f(x) = 1/(1 x), a = 0Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 55. sin 0.3, n = 4Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 56. cos 0.45, n = 3Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 57. e0.25, n = 4Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 58. tan 0.3, n = 2Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 59. e0.5, n = 4Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique. 60. ln 1.04, n = 3Error bounds Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. 61. sin x x x3/6 on [/4, /4]54EError bounds Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. 63. ex 1 + x + x2/2 on [12,12]Error bounds Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. 64. tan x x on [/6, /6]Error bounds Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. 65. ln(1 + x) x x2/2 on [0.2, 0.2]Error bounds Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. 66. 1+x1+x/2 on [0.1, 0.1]Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 67. e0.5Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 68. sin 0.2Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 69. cos (0.25)Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 70. ln 0.85Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 71. 1.06Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 103? (The answer depends on your choice of a center.) 72. 1/0.85Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Only even powers of x appear in the Taylor polynomials for f(x) = e2x centered at 0. b. Let f(x) = x5 1. The Taylor polynomial for f of order 10 centered at 0 is f itself. c. Only even powers of x appear in the nth-order Taylor polynomial for f(x)=1+x2 centered at 0. d. Suppose f is continuous on an interval that contains a, where f has an inflection point at a. Then the second-order Taylor polynomial for f at a is linear.66EMatching functions with polynomials Match functions af with Taylor polynomials AF (all centered at 0). Give reasons for your choices. a. 1+2x A. p2(x) = 1 + 2x + 2x2 b. 11+2x B. p2(x) = 1 6x + 24x2 c. e2x C. p2(x)=1+xx22 d. 11+2x D. p2(x) = 1 2x + 4x2 e. 1(1+2x)3 E. p2(x)=1x+32x2 f. e2x F. p2(x) = 1 2x + 2x268ESmall argument approximations Consider the following common approximations when x is near zero. a. Estimate f(0.1) and give a bound on the error in the approximation. b. Estimate f(0.2) and give a bound on the error in the approximation. 77. f(x) = sin x x70E71E72ESmall argument approximations Consider the following common approximations when x is near zero. a. Estimate f(0.1) and give a bound on the error in the approximation. b. Estimate f(0.2) and give a bound on the error in the approximation. 81. f(x)=1+x1+x/2Small argument approximations Consider the following common approximations when x is near zero. a. Estimate f(0.1) and give a bound on the error in the approximation. b. Estimate f(0.2) and give a bound on the error in the approximation. 82. f(x) = ln (1 + x) x x2/2Small argument approximations Consider the following common approximations when x is near zero. a. Estimate f(0.1) and give a bound on the error in the approximation. b. Estimate f(0.2) and give a bound on the error in the approximation. 83. f(x) = ex 1 + x76E77E78E79E80E81E82ETangent line is p1 Let f be differentiable at x = a. a. Find the equation of the line tangent to the curve y = f(x) at (a, f(a)). b. Verify that the Taylor polynomial p1 centered at a describes the tangent line found in part (a).Local extreme points and inflection points Suppose f has continuous first and second derivatives at a. a. Show that if f has a local maximum at a, then the Taylor polynomial p2 centered at a also has a local maximum at a. b. Show that if f has a local minimum at a, then the Taylor polynomial p2 centered at a also has a local minimum at a. c. Is it true that if f has an inflection point at a, then the Taylor polynomial p2 centered at a also has an inflection point at a? d. Are the converses in parts (a) and (b) true? If p2 has a local extreme point at a, does f have the same type of point at a?85EApproximating In x Let f(x) = ln x and let pn and qn be the nth-order Taylor polynomials for f centered at 1 and e, respectively. a. Find p3 and q3. b. Graph f, p3, and q3 on the interval (0, 4]. c. Complete the following table showing the errors in the approximations given by p3 and q3 at selected points. x |ln x p3(x)| |ln x q3(x)| 0.5 1.0 1.5 2 2.5 3 3.5 d. At which points in the table is p3 a better approximation to f than q3? Explain your observations.Approximating square roots Let p1 and q1 be the first-order Taylor polynomials for f(x)=x centered at 36 and 49, respectively. a. Find p1 and q1. b. Complete the following table showing the errors when using p1 and q1 to approximate f(x) at x = 37, 39, 41, 43, 45, and 47. Use a calculator to obtain an exact value of f(x). x xp1(x) xq1(x) 37 39 41 43 45 47 c. At which points in the table is p1 a better approximation to f than q1? Explain this result.A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivatives at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x) = sin x on the interval [0, ]. a. Write the (quadratic) Taylor polynomial p2 for f centered at 2. b. Now consider a quadratic interpolating polynomial q(x) = ax2 + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied: q(0)=f(0),q(2)=f(2),andq()=f(). Show that q(x)=42x2+4x. c. Graph f, p2, and q on [0, ]. d. Find the error in approximating f(x) = sin x at the points 4,2,34, and using p2, and q. e. Which function, p2 or q, is a better approximation to f on [0, ]? Explain.By substituting x = 0 in the power series for g, evaluate g(0) for the function in Figure 11.13. Figure 11.13What are the radius and interval of convergence of the geometric series xk?Use the result of Example 4 to write a series representation for ln12=ln2. Example 4 Differentiating and Integrating Power Series Consider the geometric series f(x)=11x=k=0xk=1+x+x2+x3+,for|x|1. a. Differentiate this series term by term to find the power series for f and identify the function it represents. b. Integrate this series term by term and identify the function it represents.4QCWrite the first four terms of a power series with coefficients c0, c1, c2, and c3 centered at 0.Is k=0(5x20)k a power series? If so, find the center a of the power series and state a formula for the coefficients ck of the power series.What tests are used to determine the radius of convergence of a power series?Is k=0x2ka power series? If so, find the center a of the power series and state a formula for the coefficients ck of the power series.Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.Suppose a power series converges if |x 3| 4 and diverges if |x 3| 4. Determine the radius and interval of convergence.Suppose a power series converges if |4x 8| 40 and diverges if |4x 8| 40. Determine the radius and interval of convergence.Suppose the power series k=0ck(xa)k has an interval of convergence of (3, 7]. Find the center a and the radius of convergence r.Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 9. k=0(2x)kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 10. k=0(x1)kk!Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 11. k=1(kx)kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 12. k=0k!(x10)kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 13. k=1sink(1k)xkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 14. k=22k(x3)kkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 15. k=0(x3)kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 16. k=0(1)kxk5kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 17. k=1xkkkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 18. k=1(1)kxkkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 19. k=0xk2k+1Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 20. k=0(2x)kk!Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 21. x21!+x42!x63!+x84!Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 22. xx34+x59x716+Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 23. k=1(1)k+1(x1)kkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 24. k=0(1)k(x4)k2kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 25. k=0(4x1)kk2+4Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 26. k=1(3x+2)kkRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 27. k=0k10(2x4)k10k9-36. Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
28.
Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 29. k=0k2x2kk!Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 30. k=0k(x1)kRadius and interval of convergence Determine the radius and interval of convergence of the following power series. 31. k=1x2k+13k1Radius and interval of convergence Determine the radius and interval of convergence of the following power series. 32. k=0(x10)2kRadius of interval of convergence Determine the radius and interval of convergence of the following power series. 33. k=1(x1)kkk(k+1)kRadius of interval of convergence Determine the radius and interval of convergence of the following power series. 34. k=0(2)k(x+3)k3k+1Radius of interval of convergence Determine the radius and interval of convergence of the following power series. 35. k=0k20xk(2k+1)!Radius of interval of convergence Determine the radius and interval of convergence of the following power series. 36. k=0(1)kx3k27kRadius of convergence Find the radius of convergence for the following power series. 37. k=1k!xkkkRadius of convergence Find the radius of convergence for the following power series. 38. k=1(1+1k)k2xkRadius of convergence Find the radius of convergence for the following power series. 39. k=1(kk+4)k2xkRadius of convergence Find the radius of convergence for the following power series. 40. k=1(1cos12k)xkCombining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 29. f(3x)=113xCombining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 30. g(x)=x31xCombining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 31. h(x)=2x31xCombining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 32. f(x)3=11x3Combining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 33. p(x)=4x121xCombining power series Use the geometric series f(x)=11x=k=0xk,for|x|1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series. 34. f(4x)=11+4xCombining power series Use the power series representation f(x)=ln(1x)=k=1xkk,for1x1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 35. f(3x) = ln (1 3x)Combining power series Use the power series representation f(x)=ln(1x)=k=1xkk,for1x1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 36. g(x) = x3 ln (1 x)Combining power series Use the power series representation f(x)=ln(1x)=k=1xkk,for1x1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 39. p(x) = 2x6 ln (1 x)Combining power series Use the power series representation f(x)=ln(1x)=k=1xkk,for1x1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 38. f(x3) = ln (1 x3)Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 41. g(x)=2(12x)2 using f(x)=112xDifferentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 42. g(x)=1(1x)3 using f(x)=11xDifferentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 43. g(x)=1(1x)4 using f(x)=11xDifferentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 44. g(x)=x(1+x2)2 using f(x)=11+x2Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 45. g(x) = ln (1 3x) using f(x)=113xDifferentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. 46. g(x) = ln (1 + x2) using f(x)=x1+x2Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the reselling series. 57. f(x)=2x(1+x2)2Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 48. f(x)=11x4Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 49. f(x)=33+xFunctions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 50. f(x)=ln1x2Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 51. f(x)=ln4x2Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 52. f(x)=tan1(4x2)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The internal of convergence of the power series ck(x3)k could be (2, 8). b. The series k0(2x)k converges on the interval 12x12. c. If f(x)=k=0ckxk on the interval |x| 1, then f(x2)=ckx2k on the interval |x| 1. d. If f(x)k=0ckxk0, for all x on an interval (-a, a), then ck = 0, for all k.Scaling power series If the power series f(x)=ckxk has an interval of convergence of |x| R, what is the interval of convergence of the power series for f(a), where a 0 is a real number?Shifting power series If the power series f(x)=ckxk has an interval of convergence of |x| R, what is the interval of convergence of the power series for f(x a), where a 0 is a real number?A useful substitution Replace x with x 1 in the series ln(1+x)=k=1(1)k+1xkk to obtain a power series for ln x centered at x = 1. What is the interval of convergence for the new power series?Series to functions Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) 63. k=0(x2)kSeries to functions Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) 64. k=1x2k4k69ESeries to functions Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) 66. k=1(x2)k32kSeries to functions Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) 67. k=0(x213)kExponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is ex=k=0xkk!, for x . Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resuming series. 74. f(x) = e2xExponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is ex=k=0xkk!, for x . Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resuming series. 74. f(x) = x3x74E75ERemainders Let f(x)=k=0xk=11xandSn(x)=k=0n1xk. The remainder in truncating the power series after n terms is Rn(x) = f(x) Sn(x), which depends on x. a. Show that Rn(x) = xn/(1 x). b. Graph the remainder function on the interval |x| 1 for n = 1, 2, 3. Discuss and interpret the graph. Where on the interval is |Rn(x)| largest? Smallest? c. For fixed n, minimize |Rn(x)| with respect to x. Does the result agree with the observations in part (b)? d. Let N(x) be the number of terms required to reduce |Rn(x)| to less than 106. Graph the function N(x) on the interval |x| 1. Discuss and interpret the graph.77EInverse sine Given the power series 11x2=1+12x2+1324x4+135246x6+, for 1 x 1, find the power series for f(x) = sin1 x centered at 0.Verify that if the Taylor series for f centered at a is evaluated at x = a, then the Taylor series equals f(a).Based on Example 1b, what is the Taylor series for f(x) = (1 + x)1?3QC4QC5QC6QCHow are the Taylor polynomials for a function f centered at a related to the Taylor series for the function f centered at a?What conditions must be satisfied by a function f to have a Taylor series centered at a?Find a Taylor series for f centered at 2 given that f(k)(2) = 1, for all nonnegative integers k.Find a Taylor series for f centered at 0 given that f(k)(0) (k + 1), for all nonnegative integers k.Suppose you know the Maclaurin series for f and that it converges to f(x) for |x| 1. How do you find the Maclaurin series for f(x2) and where does it converge?For what values of p does the Taylor series for f(x) = (1 + x)p centered at 0 terminate?