Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of x2 - x - 5 between 2 and 3
> Sum an appropriate infinite series to find the rational number whose decimal expansion is given. 4.011̅0̅1̅1̅ (= 4 + .011011)
> Sum an appropriate infinite series to find the rational number whose decimal expansion is given. .15151̅5̅
> Sum an appropriate infinite series to find the rational number whose decimal expansion is given. .222̅
> Sum an appropriate infinite series to find the rational number whose decimal expansion is given. .17317̅3̅
> Sum an appropriate infinite series to find the rational number whose decimal expansion is given. .27272̅7̅
> Determine the sums of the following geometric series when they are convergent. 53/3 – 55/34 + 57/37 - 59/310 + 511/313 - …
> Determine the sums of the following geometric series when they are convergent. 5 + 4 + 3.2 + 2.56 + 2.048 + …
> Determine the sums of the following geometric series when they are convergent. 32/25 + 34/28 + 36/211 + 38/214 + 310/217 + …
> Sketch the graphs of f (x) = sin x and its first three Taylor polynomials at x = 0.
> Determine the sums of the following geometric series when they are convergent. 2/54 - 24/55 + 27/56 - 210/57 + 213/58 - …
> Determine the sums of the following geometric series when they are convergent. 6 - 1.2 + .24 - .048 + .0096 - …
> Determine the sums of the following geometric series when they are convergent. 3 - 32/7 + 33/72 - 34/73 + 35/74 - …
> Determine the sums of the following geometric series when they are convergent. 1/32 - 1/33 + 1/34 - 1/35 + 1/36 - …
> Determine the sums of the following geometric series when they are convergent. 1/5 + 1/54 + 1/57 + 1/510 + 1/513 + …
> Determine the sums of the following geometric series when they are convergent. 3 + 6/5 + 12/25 + 24/125 + 48/625 + …
> Determine the sums of the following geometric series when they are convergent. 2 + 2/3 + 2/9 + 2/27 + 2/81 + …
> Determine the sums of the following geometric series when they are convergent. 1 + 1/23 + 1/26 + 1/29 + 1/212 + …
> Determine the sums of the following geometric series when they are convergent. 1 – 1/32 + 1/34 – 1/36 + 1/38 - …
> Determine the sums of the following geometric series when they are convergent. 1 + ¾ + (3/4)2 + (3/4)3+ (3/4)4+ …
> 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: 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
> 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)?
