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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Calculus Volume 2

In the following exercises, use the ratio test to determine the radius of convergence of each series. 30. n=12 3n (n!)3(3n)!xn In the following exercises, use the ratio test to determine the radius of convergence of each series. 31. n=1n!nnxn In the following exercises, use the ratio test to determine the radius of convergence of each series. 32. n=1(2n)!n 2nxn In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 33. f(x)=1x;a=1(Hint:1x=11(1x))In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 34. f(x)=11x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 35. f(x)=x1x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 36. f(x)=11+x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 37. f(x)=x21+x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 38. f(x)=12x;a=1 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 39. f(x)=112x;a=0.In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 40. f(x)=114x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 41. f(x)=x214x2;a=0 In the following exercises, given that 11x=n=0xn with convergence in ( 1, 1), find the power series for each function with the given center a, and identify its interval of convergence. 42. f(x)=x254x+x2;a=2 Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 43. Explain why, if |an|1/nr0, then |anxn|1/n|x|r1 whenever |x|1r and, therefore, the radius of convergence of n=1anxnisR=1r .Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 44. n=1xnnn Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 45. k=1( k1 2k+3 )kxk Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 46. k=1( 2 k 2 1 k 2 +3 )kxk Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 47. n=1an=(n 1/n1)nxn Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 48. Suppose that p(x)=n=0anxn such that an=0 if n is even. Explain why p(x) = p(x).Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 49. Suppose that p(x)=n=0anxn such that an=0 if n is odd. Explain why p(x)=p(x) .Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 50. Suppose that p(x)=n=0anxnconverges on (1, 1). Find the interval of convergence of p(Ax).Use the next exercise to find the radius of convergence of the given series in the subsequent exercises. 51. Suppose that p(x)=n=0anxnconverges on (1, 1). Find the interval of convergence of p(2x 1).In the following exercises, suppose that p(x)= n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 52. n=0anx2n In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 53. n=0a2nx2n In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 54. n=0a2nxn (Hint:x=x2)In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 55. n=0an2xn2 (Hint:Letbk=akifk=n2 for some n otherwise bk=0) .In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 56. Suppose that p(x)is a polynomial of degree N. Find the radius and interval of convergence of n=1p(n)xn .In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 57. [T] Plot the graphs of 11x and of the partial sums SN=n=0Nxn for n=10,20,30 on the interval [-0.99, 0.99]. Comment on the approximation of 11x by SN near x=1 and near x=1 as N increases.In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 58. [T] Plot the graphs of In(1x) and of the partial sums SN=n=1Nxnn for n = 10, 50, 100 on the interval [-0.99, 0.99]. Comment on the behavior of the sums near x=1 and near x=1 as N increases.In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 59. [T] Plot the graphs of the partial sums Sn=n=1Nxnn2 for n = 10, 50, 100 on the interval [-0.99, 0.99]. Comments on the behavior of the sums near x=1 and near x=1 as N increases.In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 60. [T] Plot the graphs of the partial sums SN=n=1NSinnxn for n =10, 50, 100. Comments on the behavior of the sums near x=1 and near x=1 as N increases.In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 61. [T] Plot the graphs of the partial sums SN=n=0N(1)nx2n+1(2n+1)! For n =3, 5, 10 on the interval [2,2] . Comment on the how these plots approximate sinx as N increases.In the following exercises, suppose that p(x)=n=0anxn Satisfies limn an+1an=1 where an0 for each n . State whether each series converges on the full interval ( 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate. 62. [T] Plot the graphs of the partial sums SN=n=0N(1)nx2n(2n)! for n = 3, 5, 10 on the interval [2,2] . Comment on the how these plots approximate cosx as N increases.If f(x)=n=0xnn! and g(x)=n=0(1)nxnn! , find the power series of 12(f(x)+g(x)) and 12(f(x)g(x)) .If C(x)=n=0x2n(2n)! and S(x)=n=0x2n+1(2n+1)! find the power series of C(x)+S(x) and of C(x)S(x) .In the following exercises, use partial fractions to find the power series of each function. 65. 4(x3)(x+1)In the following exercises, use partial fractions to find the power series of each function. 66. 3(x+2)(x1)In the following exercises, use partial fractions to find the power series of each function. 68. 30(x2+1)(x29)In the following exercises, use partial fractions to find the power series of each function. 68. 30(x2+1)(x29)In the following exercises, express each series as a rational function. 69. n=11xn In the following exercises, express each series as a rational function. 70. n=11x 2n In the following exercises, express each series as a rational function. 71. n=11 (x3) 2n1 In the following exercises, express each series as a rational function. 72. n=11 (x3) 2n11(x2)2n1 The following exercises explore applications of annuities. 73. Calculate the present values P of an annuity in which $10,000 is to be paid out annually for a period of 20 years, assuming interest rates of r = 0.03. r = 0.05. and r = 0.07.The following exercises explore applications of annuities. 74. Calculate the present values P of annuities in which $9,000 is to be paid out annually perpetually, assuming interest rates of r = 0.03, r = 0.05 and r = 0.07.The following exercises explore applications of annuities. 75. Calculate the annual payouts C to be given for 20 years on annuities having present value $100,000 assuming respective interest rates of r = 0.03. r = 0.05. and r = 0.07.The following exercises explore applications of annuities. 76. Calculate the annual payouts C to be given perpetually on annuities having present value $100,000 assuming respective interest rates of r = 0.03, r = 0.05. and r = 0.07.The following exercises explore applications of annuities. 77. Suppose that an annuity has a present value P = 1 million dollars. What interest rate r would allow for perpetual annual payouts of $50,000?The following exercises explore applications of annuities. 78. Suppose that an annuity has a present value P = 10 million dollars. What interest rate r would allow for perpetual annual payouts of $100,000?In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. 79. x+x2x3+x4+x5+x6+... (Hint: Group powers x3k,x3k1, and x3k2 .)In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. 80. x+x2x3x4+x5+x6x7x8+...(Hint: Group powers x4k, x4k 1, etc.)In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. 81. xx2x3+x4x5x6+x7... (Hint: Group power x3kx3k1 and x3k2.)In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. 82. x2+x24x38+x416+x532+x664+... (Hint: Group powers (x2)3k,(x2)3k1 , and (x2)3k2 .)In the following exercises, find the power series of f(x)g(x) given f and g as defined. 83. f(x)=2n=0xn,g(x)=n=0nxn In the following exercises, find the power series of f(x)g(x) given f and g as defined. 84. . f(x)=n=1xn,g(x)=n=112xn Express the coefficients of f(x)g(x) in terms of Hn=k=1n1k In the following exercises, find the power series of f(x)g(x) given f and g as defined. 85. f(x)=g(x)=n=1( x2 )n In the following exercises, find the power series of f(x)g(x) given f and g as defined. 86. f(x)=g(x)=n=1nxn In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. 87. f(x)=11x=n=0(1)nxn In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. 88. f(x)=11x2=n=0x2n In the following exercises, integrate the given series expansion of f term-by-term from zero to x to obtain the corresponding series expansion for the indefinite integral of f. 89. f(x)=2x( 1+x2 )2=n=1(1)n(2n)x2n1 In the following exercises, integrate the given series expansion of f term-by-term from zero to x to obtain the corresponding series expansion for the indefinite integral of f. 90. f(x)=2x1+x2=2n=0(1)n(2n)x2n+1 In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series. 91. Evaluate n=1n2nas f(12) where f(x)=n=0xn.In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series. 92. Evaluate n=1n3n as f(13) where. f(x)=n=0xn.In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series. 93. Evaluate n=2n(n1)2n as f(12) where f(x)=n=0xn.In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series. 94. Evaluate n=0 (1)n1+n as 01f(t)dtwhere f(x)=n=0(1)nx2n=11+x2.In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 95. f(x) = In x centered at x = 1 (Hint x = 1 (1 x))In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 96. In(1x) at x =0 In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 97. In (1 x2) at ( x = 0 In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 98. f(x)=2x(1x2)2 at x = 0 In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 99. f(x) = tan-1(x2) at x = 0 In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 100. f(x) = 1n(1 + x2) at x= 0 In the following exercises, given that 11x=n=0xn use term-by-term differentiation or integration to find power series for each function centered at the given point. 101. f(x)=0x1ntdt where 1n(x)=n=1(1)n1(x1)nn T] Evaluate the power series expansion ln(1 + x) = (1+x)=n=1(1)n1xnn at x = 1 to show that In (2) is the sum of the alternating harmonic series. Use the alternating series test to determine how many terms of the sum are needed to estimate In (2) accurate to within 0.001, and find such an approximation.[T] Subtract the infinite series of 1n(1 x) from in (1 + x) to get a power series for in (1+x1x) . Evaluate at x = 13 . What is the smallest N such that the Nth partial sum of this series approximates in (2) with an error less than 0.001?In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 104. n=0(xkx 2k1)In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 105. n=1x 3k6k In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 106. n=1(1+x2) using y=11+x2 In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 107. n=12kx using y=2x In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 108. Show that, up to powers x3and y3, E(x) = n=0xnn! satisfies E(x + y) = E(x)E(y).In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 109. Differentiate the series Ex=n=0xnn! term-by-term to show that E(x) is equal to its derivative.In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 110. Show that if f(x)=n=0anxn is a sum of even powers, that is, an= 0 if n is odd, then F=0xf(t)dtis a sum of odd powers, while if f is a sum of odd powers, then F is a sum of even powers.In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 111. [T] Suppose that the coefficients anof the series n=0anxn are defined by the recurrence relation an=an1n+an2n(n1) . For a0=0 and a1= l, compute and plot the sums SN=n=0anxn for N = 2, 3, 4, 5 on [1, 1].In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 112. [T] Suppose that the coefficients anof the series n=0anxn are defined by the recurrence relation an=an1n+an2n(n1) . For a0= 1 and a1=0, compute and plot the sums SN=n=0Nanxn for N=2, 3, 4, 5 on [1, 1].In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 113. [T] Given the power series expansion 1n(1+x)=n=1(1)n1xnn , determine how terms N of the sum evaluated at x = 1/2 are needed to approximate 1n(2) accurate to within 1/1000. Evaluate the corresponding partial sum n=1(1)n1xnn In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 114. [T] Given the power series expansion tan1(x)=k=0(1)kx 2k+12k+1 , use the alternating series test to determine how many terms N of the sum evaluated at x = 1 are needed to approximate tan-1(1) = 4 accurate to within 1/1000. Evaluate the corresponding partial stun k=0(1)kx 2k+12k+1 .In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 115. [T] Recall that tan1(13)=6 . Assuming an exact value of (13) , estimate 6 by evaluating partial sums SN(13) of the power series expansion tan1(x)=k=0(1)kx 2k+12k+1at13 . What is the smallest number N such that 6SN(13) approximates accurately so within 0,001? How many term are needed for accuracy so within 0.00001?In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 1. Write the Macbunn polynomials p0(x), p1(x), P2(x). p3(x), p4(x) for ex. Evaluate P0(1), P1(1).P2(1), P3(1), P4(1) to estimate e.In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 2. Let Rn(x) denote the remainder when using pn(x) to estimate ex. Therefore, Rn(x) = exand Rn(1) = e - pn(1). Assuming that e= rs for integers r and s, evaluate R0(l), R1(1), R2(l), R3(1), R4(1).In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 3. Using the results from part 2, show that for each remainder R0(1), R1(1), R2(1), R3(1), R4(1), we can find an integer k such that kRn(1) is an integer for n = 0, 1, 2, 3, 4.In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 4. Write down the formula for the nth Maclaurin polynomial pn(x) for exand the corresponding remainder Rn(x). Show that sn!Rn(1) is an integer.In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 5. Use Taylor’s theorem to write down an explicit formula for Rn(1). Conclude that Rn(1) ? 0, and therefore. sn!Rn(l) ?0.In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 0. 6. Use Taylor’s theorem to find an estimate on Rn(1). Use this estimate combined with the result from part 5 to show that sn!Rnsen+1 . Conclude that if n is large enough then sn!Rn1 . Therefore. sn!Rn(l) is an integer with magnitude less than 1. Thus, .sn!Rn(1) = 0. But from part 5, we know that sn!Rn(l) ? 0. We have arrived at a contradiction, and consequently, the original supposition that e is rational must be false.In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 116. f(x)=1+x+x2ata=1 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 117. f(x)=1+x+x2ata=-1 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 118. f(x)=cos(2x)ata=In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 119. f(x)=sin(2x)ata=2 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 120. f(x)=xata=4 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 121. f(x)=1nxata=1 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 122. f(x)=1xata=1 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. 123. f(x)=exata=1 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 124. [T] 10 ; a = 9, n = 3 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 125. [T] (28)1/3 a = 27, n = 1 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 126. [T] sin(6); a = 2 , n = 5 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 127. [T] e2; a = 0, n = 9 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 128. [T] cos 5 ; a = 0, n = 4 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 129. [T] 1n(2); a = 1, n = 1000 In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 130. Integrate the approximation sinttt36+t5120t75040 evaluated at t to approximate 01sinttdt.In the following exercises, verify that the given choice of n in the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 . Find the value of the Taylor polynomial Pnof f at the indicated point. 131. Integrate the approximation ex1+x+x22+...+x6720evaluated at x2to approximate 01e x 2dx.In the following exercises, find the smallest value of n such that the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 on the indicated interval. 132. f(x) = sin x on j—,r, n1. a = 0 In the following exercises, find the smallest value of n such that the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 on the indicated interval. 133. f(x) = cos x on [2,2] , a = 0 In the following exercises, find the smallest value of n such that the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 on the indicated interval. 134. f(x) = e-2x Of [l, -1], a = 0 In the following exercises, find the smallest value of n such that the remainder estimate RnM(n+1)!(xa)n+1 where M is the maximum value of f(n+1)(z) on the interval between a and the indicated point, yields Rn11000 on the indicated interval. 135. f(x) = e-x on [-3, 3], a = 0 In the following exercises, the maximum of the right-hand side of the remainder estimate on R1maxf(z)2R2 on [a-R, a+R] occurs at a or a ± R. Estimate the maximum value of R such that maxf(z)2R20.1 on [a - R, a + R] by plotting this maximum as a function of R. 136. [T] exapproximated by 1 + x. a = 0 In the following exercises, the maximum of the right-hand side of the remainder estimate on R1maxf(z)2R2 on [a-R, a+R] occurs at a or a ± R. Estimate the maximum value of R such that maxf(z)2R20.1 on [a - R, a + R] by plotting this maximum as a function of R. 137. [T] sin x approximated by x, a = 0 In the following exercises, the maximum of the right-hand side of the remainder estimate on R1maxf(z)2R2 on [a-R, a+R] occurs at a or a ± R. Estimate the maximum value of R such that maxf(z)2R20.1 on [a - R, a + R] by plotting this maximum as a function of R. 138. [T] in x approximated by x - 1, a = 1 In the following exercises, the maximum of the right-hand side of the remainder estimate on R1maxf(z)2R2 on [a-R, a+R] occurs at a or a ± R. Estimate the maximum value of R such that maxf(z)2R20.1 on [a - R, a + R] by plotting this maximum as a function of R. 139. [T] cos x approximated by 1, a = 0 In the following exercises, find the Taylor series of the given function centered at the indicated point. 140. x4at a =1 In the following exercises, find the Taylor series of the given function centered at the indicated point. 141. 1+x+x2+x3at a = 1 In the following exercises, find the Taylor series of the given function centered at the indicated point. 142. sinx at a =In the following exercises, find the Taylor series of the given function centered at the indicated point. 143. cos x at a = 2 In the following exercises, find the Taylor series of the given function centered at the indicated point. 144. sin x at x= 2 In the following exercises, find the Taylor series of the given function centered at the indicated point. 145. cos x at x = 2 In the following exercises, find the Taylor series of the given function centered at the indicated point. 146. exat a = 1 In the following exercises, find the Taylor series of the given function centered at the indicated point. 147. exat a = l In the following exercises, find the Taylor series of the given function centered at the indicated point. 148. 1(x1)2 at a = 0 (Hint: Differentiate F(x)=0xcos(t)dt;f(t)=n=0(1)ntn(2n)! )In the following exercises, find the Taylor series of the given function centered at the indicated point. 149. 1(x1)3 at a = 0 In the following exercises, find the Taylor series of the given function centered at the indicated point. 150. F(x)=0xcos(t)dt; f(t)=n=0(1)ntn(2n)!at a = 0 (Note: f is the Taylor series of cos ( t )In the following exercises, compute the Taylor series of each function around x = 1. 151. f(x)=2x In the following exercises, compute the Taylor series of each function around x = 1. 152. f(x)=x3 In the following exercises, compute the Taylor series of each function around x = 1. 153. f(x)=(x1)2 In the following exercises, compute the Taylor series of each function around x = 1. 154. f(x)=1nx In the following exercises, compute the Taylor series of each function around x = 1. 155. f(x)=1x In the following exercises, compute the Taylor series of each function around x = 1. 156. f(x)=12xx2 In the following exercises, compute the Taylor series of each function around x = 1. 157. f(x)=x4x2x21 In the following exercises, compute the Taylor series of each function around x = 1. 158. f(x)=ex In the following exercises, compute the Taylor series of each function around x = 1. 159. f(x)=e2x [T] In the following exercises, identify the value of x such that the given series n=0an is the value of the Maclaurin series of f(x) at x. Approximate the value of f(x) using S10=n=010an . 160. n=01n![T] In the following exercises, identify the value of x such that the given series n=0an is the value of the Maclaurin series of f(x) at x. Approximate the value of f(x) using S10=n=010an . 161. n=02nn![T] In the following exercises, identify the value of x such that the given series n=0an is the value of the Maclaurin series of f(x) at x. Approximate the value of f(x) using S10=n=010an . 162. n=0 (1)n (2) 2n(2n)![T] In the following exercises, identify the value of x such that the given series n=0an is the value of the Maclaurin series of f(x) at x. Approximate the value of f(x) using S10=n=010an . 163. n=0 (1)n (2) 2n+1(2n+1)!The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 164. [T] Plot sin2x (S5(x))2 on [,] . Compare the maximum difference with the square of the Taylor remainder estimate for sin x.The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 165. [T] Plot cos2x (C4(x))2 on [,] . Compare the maximum difference with the square of the Taylor remainder estimate for cos x.The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 166. [T] Plot |2S5(x)C4(x) sin(2x)| on [,] .The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 167. [T] Compare S5(x)C4(x)on [1, 1] to tan x. Compare this with the Taylor remainder estimate for the approximation of tan x by x+x33+2x515 .The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 168. [T] Plot ex e4(x) where e4(x)=1+x..x22+x36+x424 on [0, 2]. Compare the maximum error with the Taylor remainder estimate.The following exercises make use of the functions S5(x)=x+x36+x5120 and C4(x)=xx22+x224 on [,] . 169. [Taylor approximations and root finding.) Recall that Newton’s method xn+1=xnf(xn)f(xn) approximates solutions of f(x) = 0 near the input x0. a. If f and g are inverse functions, explain why a solution of g(x) = a is the value f(a) of f. b. Let PN (x) be the Nth degree Maclaurin polynomial of ex. Use Newton’s method to approximate solutions of pN(x) 2 = 0 for N = 4, 5, 6. c. Explain why the approximate roots of PN(x) 2 = 0 are approximate values of In(2).In the following exercises, use the fact that if q(x)=n=1an(nc)n converges in an interval containing c, then limxcq(x)=a0 to evaluate each limit using Taylor series. 170. limx0cosx1x2 In the following exercises, use the fact that if q(x)=n=1an(nc)n converges in an interval containing c, then limxcq(x)=a0 to evaluate each limit using Taylor series. 171. limx01n(1x2)x2 In the following exercises, use the fact that if q(x)=n=1an(nc)n converges in an interval containing c, then limxcq(x)=a0 to evaluate each limit using Taylor series. 172. limx0ex2x21x4 In the following exercises, use the fact that if q(x)=n=1an(nc)n converges in an interval containing c, then limxcq(x)=a0 to evaluate each limit using Taylor series. 173. limx0+cos(x)12x In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. 174. (1 x)1/3 In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. 175. (1+ x2)-1/3 In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. 176. (1 x)1.01 In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. 177. (1 2x)2/3 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 178. x+2 at a = 0 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 179. x2+2 at a = 0 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 180. x+2 at a =1 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 181. 2xx2 at a =1 (Hint: 2xx2= 1 (x 1)2)In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 182. (x8)at a = 9 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 183. x at a = 4 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 184. x1/3 at a =27 In the following exercises, use the substitution (b+x)=(b+a)r(1+ xa b+a)r in the binomial expansion to find the Taylor series of each function with the given center. 185 x at x =9 In the following exercises, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most 1/1000. 186. [T] (15)1/4 using (16 x)1/4 In the following exercises, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most 1/1000. 187. [T] (1001)1/3 using (1000 +x)In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 188. [T] 12 using x = in (1 — x)1/2 In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 189. [T] 5=515using x=45 in (1 x)In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 190. [T] 5=13.= using x=23 in (1 x)1/2 In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 191, [T] 6using x=56in (1 x)1/x In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 192. Integrate the binomial approximation of 1x to find an approximation of 0x1tdt.In the following exercises, use the binomial approximation 1x1x2x28x3165x51287x5256 for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator. 193, [T] Recall that the graph of 1x2 is an upper semicircle of radius 1. Integrate the binomial approximation of 1x2 up ro order 8 from x = —1 to x = 1 to estimate 2 .In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 194. (1+4x)1/3;a=0 In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 195. (1+4x)4/3;a=0 In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 196. (3+2x)1/3;a=1 In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 197. (x2+6x+10)1/3;a=3 In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 198. Use (1+x)1/3=1+13x19x2+581x310243x4+... with x = 1to approximate 21/3.In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 199. Use the approximation (1+x)2/3=1+2x3x294x3817x424314x5729+... for |x|<1 to approximate 21/3 = 2.2-2/3 In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 200. Find the 25th derivative of f(x) = (1 + x2)13 at x = 0.In the following exercises, use the expansion (1+x)1/3=1+13x19x2+581x310243x4+... to write the first five terms (not necessarily a quartic polynomial) of each expression. 201. Find the 99 th derivative of f(x) = (1 + x)25.In the following exercises, find the Maclaurin series of each function. 202. f(x)=xe2x In the following exercises, find the Maclaurin series of each function. 203. f(x)=2x In the following exercises, find the Maclaurin series of each function. 204. f(x)=sinxx In the following exercises, find the Maclaurin series of each function. 205. f(x)=sin(x)x,(x0),In the following exercises, find the Maclaurin series of each function. 206. f(x)=sin(x2)In the following exercises, find the Maclaurin series of each function. 207. f(x)=ex3 In the following exercises, find the Maclaurin series of each function. 208. f(x)=cos2x using the identity cos2x=12+12cos(2x) .In the following exercises, find the Maclaurin series of each function. 209. f(x)=sin2x using the identity sin2x=1212cos(2x) .In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 210. F(x)=0xet2dt;f(t)=et2=n=0(1)nt2nn!In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 211. F(x)=tan1x;f(t)=11+r2=n=0(1)nt2n In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 212. F(x)=tanh1x; ; f(t)=11r2=n=0t2n In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 213. F(x)=sin1x;f(t)=11r2=k=0(12k)t2kk!In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 214. F(x)=0xsinttdt;f(t)=sintt=n=0(1)nt2n(2n+1)!In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 215. F(x)=0xcos(t)dt;f(t)=n=0(1)nxn(2n)!In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 216. F(x)=0x1costt2dt;f(t)=1costt2=n=0(1)nt2n(2n+2)!In the following exercises, find the Maclaurin series of F(x)=0xf(t)dtby integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. 217. F(x)=0x1n(1+t)tdt;f(t)=n=0(1)ntnn+1 In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 218. sin(x+4)=sinxcos(4)+cosxsin(4)In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 219. f(x) = tan x In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 220. f(x) = 1n(cos x)In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 221. f(x) = excos x In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 222. f(x) = e sin x In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 223. f(x) = sec2x In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 224. f(x) = tanh x In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclain-in series of f. 225. f(x) = tanxx (see expansion for tan x)In the following exercises, find the radius of convergence of the Maclaurin series of each function. 226. 1n(1 + x)In the following exercises, find the radius of convergence of the Maclaurin series of each function. 227. 11+x2 In the following exercises, find the radius of convergence of the Maclaurin series of each function. 228. tan-1 x In the following exercises, find the radius of convergence of the Maclaurin series of each function. 229. In(1 + x2)In the following exercises, find the radius of convergence of the Maclaurin series of each function. 230. Find the Madaurin series of sinh x=exex2 In the following exercises, find the radius of convergence of the Maclaurin series of each function. 231. Find the Maclaurin series of cosh x=exex2 In the following exercises, find the radius of convergence of the Maclaurin series of each function. 232. Differentiate term by term the Maclatirin series of sinh x and compare the result with the Maclaurin series of cosh x.233. [T] Let Sn(s)=k=0n(1)kx 2k+1(2k+1)! and Cn(x)=n=0n(1)kx 2k(2k)! denote the respective Maclaurin polynomials of degree 2n + 1 of sin x and degree 2n of cos x. Plot the errors Sn(x)Cn(x)tanx for n = 1, …5 and compare them to x+x33+2x515+17x7315tanx on (4,4) .Use the identity 2 sin x cos x = sin (2x) to find the power series expansion of sin2x at x = 0. (Hint: Integrate the Maclaurin series of sin (2x) term by term.)If y=n=0anxn , find the power series expansions of xy' and xy".[T] Suppose that y=k=0akxk satisfies y'=-2xy and y(0) = 0. Show that a2k+1= 0 for all k and that a2k+2=a2kk+1 . Plot the partial sum S20of y on the interval [4, 4].[T] Suppose that a set of standardized test scores is normally distributed with mean = 100 and standard deviation = 10. Set up an integral that represents the probability that a test score will be between 90 and 110 and use the integral of the degree 10 Maclaurin polynomial of 12ex2/2 to estimate this probability.[T] Suppose that a set of standardized test scores is normally distributed with mean = 100 and standard deviation = 10. Set up an integral that represents the probability , that a test score will be between 70 and 130 and use the integral of the degree 50 Maclaurin polynomial of 12ex2/2 to estimate this probability.[T] Suppose that n=0anxn converges to a function f(x) such that f(0) = 1, f' (0) = 0, and f" (x) =-f(x). Find a formula for anand plot the partial sum SNfor N = 20 On [5, 5].[T] Suppose that n=0anxn converges to a function f(x) such that f(0) = 0, f' (0) = 1, and f" (x) = f(x). Find a formula for anand plot the partial sum SNfor N = 10 on [5, 5].Suppose that n=0anxn converges to a function y such that y" y' + y = 0 where y(0) = 1 and y' (0) = 0. Find a formula that relates an+ 2’ an+ 1’ and anand compute a0, ..., a5.Suppose that n=0anxnconverges to a function y such that yn y' + y = 0 where y(0) = 0 and y(0) = 1. Find a formula that relates an+2, an+1 and anand compute a1, .... a5. The error in approximating the integral abf(t)dt by that of a Taylor approximation abPn(t)dtis at most abRn(t)dt. In the following exercises, the Taylor remainder estimate RnM(n+1)!xan+1 guarantees that the integral of the Taylor polynomial of the given order approximates the integral of f with an error less than 110 . Evaluate the integral of the appropriate Taylor polynomial and verify that it approximates the CAS value with an error less than 1100 . Compare the accuracy of the polynomial integral estimate with the remainder estimate.[T] 0sinttdt;Ps=1 x 23!+ x 45!+ x 67!+ x 89! may assume that the absolute value of the ninth derivative of sintt is bounded by 0. 1.)[T] t;P11=1x2+x42+x63!+....x2211! May assume that the absolute val;ue of the 23rd derivative of ex2 is less than 2 × 1014.The following exercises deal with Fresnel integrals. 245. The Fresnel integrals are defined by C(x)=0xcos(t2)dt and S(x)=0xsin(t2)dt . Compute the power series of C(x) and S(x) and plot the sums CN(x) and SN(x) of the first N = 50 nonzero terms on [0. 2 ].The following exercises deal with Fresnel integrals. 246. [T] The Fresnel integrals are used in design applications for roadways and railways and other applications because of the curvature properties of the curve with coordinates (C(t). S(r)). Plot the curve (C50, S50) for 0 t 2 , 247. Estimate 01/4x x 2dx by approximating 1x using the binomial approximation 1x2x28x3165x421287x5256 .The following exercises deal with Fresnel integrals. 247. Estimate 01/4x x 2dx by approximating 1x using the binomial approximation 1x2x28x3165x421287x5256 .The following exercises deal with Fresnel integrals. 248. [T] Use Newton’s approximation of the binomial 1x2 to approximate as follows. The circle centered at (12,0) with radius 12 has upper semicircle y=x1x. The sector of this circle bounded by the x-axis between x = 0 and x=12 and by the line joining (14, 34) , corresponds to 16 of the circle and has area 24 . This sector is the union of a right triangle with height 34 and base 14 and the region below the graph between x = 0 and x=14. To find the area of this region you can write y=x1x=x (binomial expansion of 1x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate .The following exercises deal with Fresnel integrals. 249. Use the approximation T2Lg(1+k24) approximate the period of a pendulum having length 10 meters and maximum angle max=6 where k=sin(max2) . Compare this with the small angle estimate T2Lg .The following exercises deal with Fresnel integrals. 250. Suppose that a pendulum is to have a period of 2 seconds and a maximum angle of max=6 . Use T2Lg(1+k24) to approximate the desired length of the pendulum. What length is predicted by the small angle estimate T2Lg ?The following exercises deal with Fresnel integrals. 251. Evaluate 0/2sin4din the approximation T=4Lg0 /2( 1+ 1 2 k 2 sin 2+ 3 8 k 4 sin 4+.....)d to obtain an improved estimate for T.The following exercises deal with Fresnel integrals. 252. [T] An equivalent formula for the period of a pendulum with amplitude max is T( max ) = 22Lg0 maxd coscos( max)where L is the pendulum length and g is the gravitational acceleration constant. When max=3 we get 1cost1/22(1+t22+t43+181t6720) . Integrate this approximation to estimate in terms of L and g. Assuming g = 9.806 meters per second squared, find an approximate length L such that T(3) = 2 seconds.True or False? In the following exercises, justify your answer with a proof or a counterexample. 253. If the radius of convergence for a power series n=0anxn is 5, then the radius of convergence for the series n=1nanxn1 is also 5.True or False? In the following exercises, justify your answer with a proof or a counterexample. 254. Power series can be used to show that the derivative of exis ex. (Hint: Recall that ex=n=01n!xn.True or False? In the following exercises, justify your answer with a proof or a counterexample. 255. For small values of x, sin x x.True or False? In the following exercises, justify your answer with a proof or a counterexample. 256. The radius of convergence for the Maclaurin series of f(x) = 3x is 3.In the following exercises, find the radius of convergence and the interval of convergence for the given series. 257. n=0n2(x1)n In the following exercises, find the radius of convergence and the interval of convergence for the given series. 258. n=0xnxn In the following exercises, find the radius of convergence and the interval of convergence for the given series. 259. n=03nxn 12n In the following exercises, find the radius of convergence and the interval of convergence for the given series. 260. n=02nen(xe)n In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series. 261. f(x)=x2x+3 In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series. 262. f(x)=8x+22x23x+1 In the following exercises, find the power series for the given function using term—by—term differentiation or integration. 263. f(x)=tan1(2x)In the following exercises, find the power series for the given function using term—by—term differentiation or integration. 264. f(x)=x(2+x2)2 In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation? 265. f(x)=x32x3+4,a=3 In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation? 266. f(x)=e1/(4x),a=4 In the following exercises, find the Maclaurin series for the given function. 267. f(x) = cos(3x)In the following exercises, find the Maclaurin series for the given function. 268. f(x) = in (x + 1)In the following exercises, find the Taylor series at the given value. 269. f(x) = sin x, a=2 In the following exercises, find the Taylor series at the given value. 270. f(x) = 3x , a = 1 In the following exercises, find the Maclaurin series for the given function. 271. f(x)=ex21 In the following exercises, find the Maclaurin series for the given function. 272. f(x)=cosxxsinx In the following exercises, find the Maclaurin series for F(x)=0xf(t)dt by integrating the Maclaurin series of f(x) term by term. 273. f(x)=sinxx In the following exercises, find the Maclaurin series for F(x)=0xf(t)dt by integrating the Maclaurin series of f(x) term by term. 274. f(x)=1ex In the following exercises, find the Maclaurin series for F(x)=0xf(t)dt by integrating the Maclaurin series of f(x) term by term. 275. eix=cosx+isinx The following exercises consider problems of annuity payments. 276. For annuities with a present value of $1 million, calculate the annual payouts given over 25 years assuming interest rates of 1% 5%, and 10%.The following exercises consider problems of annuity payments. 277. A loner winner has an annuity that has a present value of $10 million. What interest rate would they need to live on perpetual annual payments of $250.000?The following exercises consider problems of annuity payments. 278. Calculate the necessary present value of an annuity in order to support annual payouts of $15,000 given over 25 years assuming interest rates of 1%, 5%, and 10%.The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the versiera, a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, versiera, but in Latin, this same word means a "female goblin. When Agnesis book was translated into English in 1801, the translator used the term witch for the curve, instead of rope. The name witch of Agnesi has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. On the figure, label the following points, lengths, and angle: a. C is the point 011 the x-axis with the same x-Coordinate as A. b. x is the x-coordinate of P, and y is the y-coordinate of P. c. E is the point (0, a). d. F is the point on the line segment 0A such that the line segment EF is perpendicular to the line segment OA. e. b is the distance from O to F . f. c is the distance from F to A. g. d is the distance from O to B. h. is the measure of angle COA. The goal of this project is to parameterize the witch using as a parameter. To do this, write equations for x and y in terms of only .The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 2. Show that d=2asin .The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 3. Note that x=dcos . Show that x=2acot . When you do this, you will have parameterized the x—coordinate of the curve with respect to (. If you can get a similar equation for y, you will have parameterized the curve.The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 4. In terms of (, what is the angle EOA ?The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 5. Show that b+c=2acos(2) .The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 6. Show that y=2acos(2)sin .The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 7. Show that y=2asin2 . You have now parameterized the y—coordinate of the curve with respect to The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 8. Conclude that a parameterization of the given witch curve is x=2acot,y=2asin2, .The Witch of Agnesi Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versiera,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a Circle of radius a so that the points (0, 0) and (0, 2a) are points on the circle (Figure 7.12). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0, 2a). The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. 9. Use your parameterization to show that the given witch curve is the graph of the function f(x)=8a3x2+4a2 .Travels with My Ant: The Curtate and Prolate Cycloids Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 7.13). As we have discussed, we have a lot of ?exibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 7.13, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, C=(xC,yC) , is given by xC=at and yC=a . Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xCasint=atasint=a(tsint) yA=yCacost=aacost=a(1cost) . Figure 7.13 (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up—and-down motion and is called a curtate cycloid (Figure 7.14). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 7.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel. 1. What is the position of the center of the wheel after the tire has rotated through an angle of t?Travels with My Ant: The Curtate and Prolate Cycloids Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 7.13). As we have discussed, we have a lot of ?exibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 7.13, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, C=(xC,yC) , is given by xC=at and yC=a . Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xCasint=atasint=a(tsint) yA=yCacost=aacost=a(1cost) . Figure 7.13 (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up—and-down motion and is called a curtate cycloid (Figure 7.14). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 7.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel. 2. Use geometry to ?nd expressions for xCxA and for yCyA .Travels with My Ant: The Curtate and Prolate Cycloids Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 7.13). As we have discussed, we have a lot of ?exibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 7.13, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, C=(xC,yC) , is given by xC=at and yC=a . Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xCasint=atasint=a(tsint) yA=yCacost=aacost=a(1cost) . Figure 7.13 (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up—and-down motion and is called a curtate cycloid (Figure 7.14). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 7.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel. 3. On the basis of your answers to parts 1 and 2, what are the parametric equations representing the curtate cycloid? Once the ant’s head clears, he realizes that the bicyclist has made a turn, and is now traveling away from his home. So he drops off the bicycle tire and looks around. Fortunately, there is a set of train tracks nearby, headed back in the right direction. So the ant heads over to the train tracks to wait. After a while, a train goes by, heading in the right direction, and he manages to jump up and just catch the edge of the train wheel (without getting squished!) The ant is still worried about getting dizzy, but the train wheel is slippery and has no spokes to climb, so he decides to just hang on to the edge of the wheel and hope for the best. Now, train wheels have a ?ange to keep the wheel running on the tracks. So, in this case, since the ant is hanging on to the very edge of the ?ange, the distance from the center of the wheel to the ant is actually greater than the radius of the wheel (Figure 7.15). The setup here is essentially the same as when the ant climbed up the spoke on the bicycle wheel. We let b denote the distance from the center of the wheel to the ant, and we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant (Figure 7.15). When the distance from the center of the wheel to the ant is greater than the radius of the wheel, his path of motion is called a prolate cycloid. A graph of a prolate cycloid is shown in the ?gure. Figure 7.15 (a) The ant is hanging onto-the ?ange of the train wheel. (b) The new setup, now that the ant has jumped onto the train wheel. (c) The ant travels along a prolate cycloid.Travels with My Ant: The Curtate and Prolate Cycloids Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 7.13). As we have discussed, we have a lot of ?exibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 7.13, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, C=(xC,yC) , is given by xC=at and yC=a . Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xCasint=atasint=a(tsint) yA=yCacost=aacost=a(1cost) . Figure 7.13 (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up—and-down motion and is called a curtate cycloid (Figure 7.14). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 7.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel. 4. Using the same approach you used in parts 1— 3, find the parametric equations for the path of motion of the ant.Travels with My Ant: The Curtate and Prolate Cycloids Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 7.13). As we have discussed, we have a lot of ?exibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 7.13, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, C=(xC,yC) , is given by xC=at and yC=a . Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xCasint=atasint=a(tsint) yA=yCacost=aacost=a(1cost) . Figure 7.13 (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up—and-down motion and is called a curtate cycloid (Figure 7.14). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 7.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel. 5. What do you notice about your answer to part 3 and your answer to part 4? Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be real1y dizzy by the time he gets home!For the following, exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 1. x=t2+2t,y=t+1 For the following, exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 2. x=cos(t),y=sin(t),(0,2)For the following, exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 3. x=2t+4,y=t1 For the following, exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 4. x=3t,y=2t3,1.5t3 For the following exercises, eliminate the parameter and sketch the graphs. 5. x=2t2,y=t4+1 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 6. [Tl x=t2+t,y=t21 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 7. [T] x=et,y=e2t1 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 8. [T] x=3cost,y=4sint For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 9. [T] x=sect,y=cost For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 10. x=et,y=e2t+1 For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 11. x=6sin(2),y=4cos(2)For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 12. x=cos,y=2sin(2)For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 13. x=32cos,y=5+3sin For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 14. x=4+2cos,y=1+sin For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 15. x=sect,y=tant For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 16. x=In(2t),y=t2 For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 17. x=et,y=e2t For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 18. x=e2t,y=e3t For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 19. x=t3,y=3Int For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 20. x=4sec,y=3tan For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 21. x=t21,y=t2 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 22. x=1t+1,y=t1+t,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 23. x=4cos,y=3sin,t(0,2]For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 24. x=cosht,y=sinht For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 25. x=2t3,y=6t7 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 26. x=t2,y=t3 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 27. x=1+cost,y=3sint For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 28. x=t,y=2t+4 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 29. x=sect,y=tant,t32 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 30. x=2cosht,y=4sinht For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 31. x=cos(2t),y=sint

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