The third remainder for f (x) at x = 0 is R3 (x) = f (4) (c)/4! x4, where c is a number between 0 and x. Let f (x) = cos x, as in Check Your Understanding Problem 11.1. (a) Find a number M such that | f (4)(c) | ≤ M for all values of c. (b) In Check Your Understanding Problem 11.1, the error in using p3(.12) as an approximation to f (.12) = cos .12 is given by R3(.12). Show that this error does not exceed 8.64 * 10-6.
> Suppose that an investment of $500 yields returns of $100, $200, and $300 at the end of the first, second, and third months, respectively. Determine the internal rate of return on this investment.
> Determine the fourth Taylor polynomial of f (x) = ex at x = 0, and use it to estimate e0.01.
> Use the Newton–Raphson algorithm to find an approximate solution to e5-x = 10 - x.
> Use the Newton–Raphson algorithm to find an approximate solution to e-x = x2.
> Sketch the graph of y = x3 + x - 1, and use the Newton–Raphson algorithm (three repetitions) to approximate all x-intercepts.
> Sketch the graph of y = x3 + 2x + 2, and use the Newton–Raphson algorithm (three repetitions) to approximate all x-intercepts.
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of ex + 10x - 3 near x0 = 0
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of sin x + x2 - 1 near x0 = 0
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of x2 + 3x - 11 between -5 and -6
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of x2 - x - 5 between 2 and 3
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: 3√11
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: 3√6
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = √(1 – x)
> Graph the function Y1 = cos x and its second Taylor polynomial in the window ZDecimal. Find an interval of the form [- b, b] over which the Taylor polynomial is a good fit to the function. What is the greatest difference between the two functions on this
> Graph the function Y1 = ex and its fourth Taylor polynomial in the window [0, 3] by [-2, 20]. Find a number b such that graphs of the two functions appear identical on the screen for x between 0 and b. Calculate the difference between the function and it
> Repeat Exercise 31 for the function Y1 = 1/(1 – x) and its seventh Taylor polynomial. Exercise 31: Graph the function Y1 = 1/(1 – x) and its fourth Taylor polynomial in the window [-1, 1] by [-1, 5]. Find a number b such that graphs of the two functions
> Graph the function Y1 = 1/(1 – x) and its fourth Taylor polynomial in the window [-1, 1] by [-1, 5]. Find a number b such that graphs of the two functions appear identical on the screen for x between 0 and b. Calculate the difference between the function
> Let p2 (x) be the second Taylor polynomial of f (x) = ln x at x = 1, as in Exercise 22. (a) Show that | f (3)(c) | < 4 if c ≥ .8. (b) Show that the error in using p2(.8) as an approximation for ln .8 is at most 16/3 * 10-3 < .0054. Exercise 22: Use the
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = √(4x + 1)
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = cos(π - 5x)
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = 5e2x
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = e-x/2
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = sin x
> Find the fifth Taylor polynomial of x3 - 7x2 + 8 at x = 0.
> Find the fourth Taylor polynomial of (2x + 1)3/2 at x = 0.
> Find the second Taylor polynomial of x(x + 1)3/2 at x = 0.
> Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the trust fund must pay to your descendants and their heirs
> Let p2(x) be the second Taylor polynomial of f (x) = √x at x = 9, as in Exercise 21. (a) Give the second remainder for f (x) at x = 9. (b) Show that f (3)(c) ≤ 1/648 if c ≥ 9. (c) Show that the error in using p2(9.3) as an approximation for 29.3 is at mo
> Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the trust fund must pay to your descendants and their heirs
> Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the trust fund must pay to your descendants and their heirs
> Suppose that the Federal Reserve creates $100 million of new money, as in Exercise 41, and the banks lend 85% of all new money they receive. However, suppose that out of each loan, only 80% is redeposited into the banking system. Thus, whereas the first
> Suppose that the Federal Reserve (the Fed) buys $100 million of government debt obligations from private owners. This creates $100 million of new money and sets off a chain reaction because of the “fractional reserve” banking system. When the $100 millio
> Let f (x) = x - 2x3 + 4x5 - 8x7 + 16x9 - …. (a) Find the Taylor series expansion of 1f (x) dx at x = 0. (b) Find a simple formula for 1f (x) dx not involving a series.
> Let f (x) = 1 + x2 + x4 + x6 + …. (a) Find the Taylor series expansion of f ‘(x) at x = 0. (b) Find the simple formula for f ‘(x) not involving a series.
> Let f (x) = ln| sec x + tan x|. It can be shown that f ‘(0) = 1, f ‘‘(0) = 0, f ‘‘‘(0) = 1, and f (4)(0) = 0. What is the fourth Taylor polynomial of f (x) at x = 0?
> It can be shown that the sixth Taylor polynomial of f (x) = sin x2 at x = 0 is x2 – 1/6 x6. Use this fact in parts (a), (b), and (c). (a) What is the fifth Taylor polynomial of f (x) at x = 0? (b) What is f ’’’(0)? (c) Estimate the area under the graph o
> Find an infinite series that converges to ∫0 ½ (ex – 1)/x dx.
> Use the decomposition (1 + x)/(1 – x) = 1/(1 – x) + x/(1 – x) to find the Taylor series of (1 + x)/(1 – x) at x = 0.
> Let p4 (x) be the fourth Taylor polynomial of f (x) = ex at x = 0. Show that the error in using p4(.1) as an approximation for e0.1 is at most 2.5 * 10-7.
> (a) Find the Taylor series of cos 3x at x = 0. (b) Use the trigonometric identity cos3 x = ¼ (cos 3x + 3 cos x) to find the fourth Taylor polynomial of cos3 x at x = 0.
> (a) Find the Taylor series of cos 2x at x = 0, either by direct calculation or by using the known series for cos x. (b) Use the trigonometric identity sin2 x = ½ (1 - cos 2x) to find the Taylor series of sin2 x at x = 0.
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. (ex – 1)/x
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. 1/(1 - 3x)2
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. ln(1 + x3)
> Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. 1/(1 + x3)
> For what values of p is ∑k=1 ∞ 1/pk convergent?
> For what values of p is ∑k=1 ∞ 1/kp convergent?
> Determine if the given series is convergent. ∑k=0∞ k3/(k4 + 1)2
> Determine if the given series is convergent. ∑k=1∞ (ln k)/k
> Determine if the given series is convergent. ∑k=1∞ 1/3k
> Determine if the given series is convergent. ∑k=1∞ 1/k3
> Find ∑k=0∞ (3k + 5k)/7k.
> Use properties of convergent series to find a ∑k=0∞ (1 + 2k)/3k.
> Find the sum of the given infinite series if it is convergent. 1 + 1/3 + 1/2! (1/3)2 + 1/3! (1/3)3 + 1/4! (1/3)4 + …
> Find the sum of the given infinite series if it is convergent. 1 + 2 + 22/2! + 23/3! + 24/4! + …
> Find the sum of the given infinite series if it is convergent. 1/m – 1/m2 + 1/m3 – 1/m4 + 1/m5 - …, where m is a positive number
> Find the sum of the given infinite series if it is convergent. 1/(m + 1) + m/(m + 1)2 + m2/(m + 1)3 + m3/(m + 1)4 +…, where m is a positive number
> Find the sum of the given infinite series if it is convergent. 22/7 - 25/72 + 28/73 - 211/74 + 214/75 - …
> Find the sum of the given infinite series if it is convergent. 1/8 + 1/82 + 1/83 + 1/84 + 1/85 + …
> If f (x) = 2 - 6 (x - 1) + 3/2! (x - 1)2 – 5/3! (x - 1)3 + 1/4! (x - 1)4, what are f ’’(1) and f ’’’(1)?
> Find the sum of the given infinite series if it is convergent. 52/6 + 53/62 + 54/63 + 55/64 + 56/65 + …
> Find the sum of the given infinite series if it is convergent. 1 – ¾ + 9/16 – 27/64 + 81/256 - …
> Use the Newton–Raphson algorithm with n = 3 to approximate the solution of the equation e2x = 1 + e-x.
> Use the Newton–Raphson algorithm with n = 2 to approximate the zero of x2 - 3x - 2 near x0 = 4.
> (a) Use the third Taylor polynomial of ln(1 - x) at x = 0 to approximate ln 1.3 to four decimal places. (b) Find an approximate solution of the equation ex = 1.3 using the Newton–Raphson algorithm with n = 2 and x0 = 0. Express your answer to four decima
> (a) Find the second Taylor polynomial of √x at x = 9. (b) Use part (a) to estimate √8.7 to six decimal places. (c) Use the Newton–Raphson algorithm with n = 2 and x0 = 3 to approximate the solution of the equation x2 - 8.7 = 0. Express your answer to six
> Use a second Taylor polynomial at x = 0 to estimate the value of tan(.1).
> Use a second Taylor polynomial at t = 0 to estimate the area under the graph of y = -ln(cos 2t) between t = 0 and t = 1/2.
> Find the third Taylor polynomial of ex at x = 2.
> Find the third Taylor polynomial of x2 at x = 3.
> If f (x) = 3 + 4x – 5/2! x2 + 7/3! x3, what are f ’’(0) and f ’’’(0)?
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = xe3x
> Find the nth Taylor polynomial of 2/(2 – x) at x = 0.
> In what way is the nth Taylor polynomial of f (x) at x = a like f (x) at x = a?
> Define the nth Taylor polynomial of f (x) at x = a.
> Discuss the three possibilities for the radius of convergence of a Taylor series.
> Define the Taylor series of f (x) at x = 0.
> What is the sum of a convergent geometric series?
> What is a geometric series and when does it converge?
> What is meant by the sum of a convergent infinite series?
> What is a convergent infinite series? Divergent?
> What is the nth partial sum of an infinite series?
> Determine the nth Taylor polynomial of f (x) = 1/x at x = 1.
> Explain how the Newton–Raphson algorithm is used to approximate a zero of a function.
> State the remainder formula for the nth Taylor polynomial of f (x) at x = a.
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. ln(1 - 3x)
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. 1/(2x + 3)
> Let Rn(x) be the nth remainder of f (x) = ex at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ e|x| * |x |n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This shows that the Taylor series for ex converges to ex fo
> Let Rn(x) be the nth remainder of f (x) = cos x at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ |x|n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This shows that the Taylor series for cos x converges to cos x f
> If k is a positive constant, show that x2e-kx approaches 0 as x→∞.
> Show that ex > x3/6 for x > 0, and from this, deduce that x2 e-x approaches 0 as x→∞.
> Let k be a positive constant. (a) Show that ekx > k2x2/2, for x > 0. (b) Deduce that e-kx 0. (c) Show that x e-kx approaches 0 as x→∞.
> (a) Use the Taylor series for ex at x = 0 to show that ex > x2/2 for x > 0. (b) Deduce that e-x < 2/x2 for x > 0. (c) Show that xe-x approaches 0 as x → ∞.
> Determine all Taylor polynomials of f (x) = x4 + x + 1 at x = 2.
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 x ex3 dx
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 e-x2 dx
> Find an infinite series that converges to the value of the given definite integral. ∫0 1 sin x2 dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫1/(1 + x3) dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫x ex3 dx
> Find the Taylor series expansion at x = 0 of the given antiderivative. ∫e-x2 dx
