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 c1 dollars (total) at the end of the first year, c2 dollars at the end of the second year, c3 dollars at the end of the third year, and so on, forever. The amount that must be in the trust fund initially to make the kth payment is cke-0.08k, the present value of the amount to be paid in k years. So the life insurance policy should pay a total of ∑k=1∞ cke-0.08k dollars into the trust fund to provide for all the payments. How large must the insurance policy be if ck = 10,000(.9)k for all k?
> Sketch the graphs of f (x) = 1/(1 – x) and its first three Taylor polynomials at x = 0.
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: √7
> Use three repetitions of the Newton–Raphson algorithm to approximate the following: √5
> Graph the function f (x) = x2 /(1 + x2), [-2, 2] by [-.5, 1]. The function has 0 as a zero. By looking at the graph, guess at a value of x0 for which x1 will be exactly 0 when the Newton– Raphson algorithm is invoked. Then, test your guess by carrying ou
> Draw the graph of f (x) = x4 - 2x2, [-2, 2] by [-2, 2]. The function has zeros at x = -12, x = 0, and x = 12. By looking at the graph, guess which zero will be approached when you apply the Newton–Raphson algorithm to each of the following initial approx
> Apply the Newton–Raphson algorithm to the function f (x) = x3 - 5x with x0 = 1. After observing the behavior, graph the function along with the tangent lines at x = 1 and x = -1, and explain geometrically what is happening.
> The functions f (x) = x2 - 4 and g (x) = (x - 2)2 both have a zero at x = 2. Apply the Newton–Raphson algorithm to each function with x0 = 3, and determine the value of n for which xn appears on the screen as exactly 2. Graph the two functions and explai
> Apply the Newton–Raphson algorithm to the function whose graph is drawn in Fig. 10(b). Use x0 = 1. Figure 10:
> Apply the Newton–Raphson algorithm to the function f (x) = x1/3 whose graph is drawn in Fig. 10(a). Use x0 = 1. Figure 10:
> What happens when the first approximation, x0, is actually a zero of f (x)?
> What special occurrence takes place when the Newton–Raphson algorithm is applied to the linear function f (x) = mx + b with m ≠ 0?
> Determine the fourth Taylor polynomial of f (x) = ln(1 - x) at x = 0, and use it to estimate ln(.9).
> Figure 9 contains the graph of the function f (x) = x3 - 12x. The function has zeros at x = - √12, 0, and √12. Which zero of f (x) will be approximated by the Newton–Raphson method starting with x0 =
> Figure 8 contains the graph of the function f (x) = x2 - 2. The function has zeros at x = √2 and x = - √2. When the Newton–Raphson algorithm is applied to find a zero, what values of x0 lead to the ze
> Suppose that the graph of the function f (x) has slope -2 at the point (1, 2). If the Newton–Raphson algorithm is used to find a root of f (x) = 0 with the initial guess x0 = 1, what is x1?
> Suppose that the line y = 4x + 5 is tangent to the graph of the function f (x) at x = 3. If the Newton–Raphson algorithm is used to find a root of f (x) = 0 with the initial guess x0 = 3, what is x1?
> Redo Exercise 17 with x0 = 1. Exercise 17: A function f (x) has the graph given in Fig. 7. Let x1 and x2 be the estimates of a root of f (x) obtained by applying the Newton–Raphson algorithm using an initial approximation of x0 = 5. Dr
> A function f (x) has the graph given in Fig. 7. Let x1 and x2 be the estimates of a root of f (x) obtained by applying the Newton–Raphson algorithm using an initial approximation of x0 = 5. Draw the appropriate tangent lines and estimat
> A mortgage of $100,050 is repaid in 240 monthly payments of $900. Determine the monthly rate of interest.
> A $663 flat-screen TV is purchased with a down payment of $100 and a loan of $563 to be repaid in five monthly installments of $116. Determine the monthly rate of interest on the loan.
> An investor buys a bond for $1000. She receives $10 at the end of each month for 2 months and then sells the bond at the end of the second month for $1040. Determine the internal rate of return on this investment.
> 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 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
> 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
> 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?
