2.99 See Answer

Question:


(a). Find the Taylor polynomials up to degree 6 for f (x) = cos x centered at a = 0. Graph f and these polynomials on a common screen.
(b). Evaluate f and these polynomials at x = π/4, π/2, and π.
(c). Comment on how the Taylor polynomials converge to f (x).


> Evaluate the indefinite integral as an infinite series. f ex – 1/x, dx

> Evaluate the indefinite integral as an infinite series. f x cos (x3) dx

> Find the area of the region enclosed by one loop of the curve. r=1+ 2 sin e (inner loop)

> For the limit illustrate Definition 1 by finding values of that correspond to e = 0.5 and e = 0.1 e* - 1 lim - 1

> Find a power series representation for the function and determine the interval of convergence. x? f(x) = .3 a - x .3

> (a). Use the binomial series to expand 1/ √1 - x2. (b). Use part (a) to find the Maclaurin series for sin-1 x.

> Use the Maclaurin series for sin x to compute si 30 correct to five decimal places.

> Use the Maclaurin series for ex to calculate e-0.2 correct to five decimal places.

> Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? f(x) = In(1 + x²)

> If the radius of convergence of the power series ∑∞n=0cnxn is 10, what is the radius of convergence of the series ∑∞n=0ncnxn-1? Why?

> Let fn(x) = (sin nx)/n2. Show that the series ∑fn(x) converges for all values of x but the series of derivatives ∑fn'(x) diverges when x = 2nπ, an integer. For what values of x does the series ∑fn"(x) converge?

> Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? f(x) = cos(x²)

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. sin x if x+0 f(x) = if x = 0

> Find the area of the region enclosed by one loop of the curve. r= 4 sin 30

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. .2 f(x) = /2 + x

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = - V4 + x? %3D

> The resistivity of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (Ωm). The resistivity of a given metal depends on the temperature according to the equation where is the temperature i

> An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and -q and are located at a distance d from each other, then the electric field E at the point P in the figure is E = q/D2 –

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) — е* + 2е- e

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = e + e2*

> Suppose you know that and the Taylor series of f centered at 4 converges to f (x) for all in the interval of convergence. Show that the fifth degree Taylor polynomial approximates f (5) with error less than 0.0002. (-1)*n! f®(4) 3"(n 1)

> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = sin 7x

> Use the binomial series to expand the function as a power series. State the radius of convergence. (1 – x)-/3

> Use the Alternating Series Estimation Theorem or Taylor&acirc;&#128;&#153;s Inequality to estimate the range of values of for which the given approximation is accurate to within the stated error. Check your answer graphically. x (| error |< 6 :0.01)

> Find the area of the region enclosed by one loop of the curve. r= sin 20

> Use the binomial series to expand the function as a power series. State the radius of convergence. 1 (1 + x)* 4

> Use the binomial series to expand the function as a power series. State the radius of convergence. + x

> Use the information from Exercise 14 to estimate sin 380 correct to five decimal places. Exercise 14: f(x) = sin x, a = "/6, n= 4, 0<x</3

> Use the information from Exercise 5 to estimate cos 800 correct to five decimal places. Exercise 5: Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. = cos x, a = T/2 1 = 7/2

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> Graph the curve and find the area that it encloses. r= 2 sin e + 3 sin 90

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (x) &acirc;&#137;&#136; Tn (x) when x lies in the given interval. (c). Check your resu

> Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn (x) &acirc;&#134;&#146; 0.] Also find the associated radius of convergence. f(x) = xe хе

> Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn (x) &acirc;&#134;&#146; 0.] Also find the associated radius of convergence. f(x) = e5=

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = tan-'x, a = 1

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) %3D хе -2:, а %3D 0 = xe a = 0

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) In x a = 1

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. = cos x, a = T/2 1 = 7/2

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = x + e, a=0

> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = 1/x, a = 2

> Graph the curve and find the area that it encloses.

> (a). Find the Taylor polynomials up to degree 3 for f (x) = 1/x centered at a = 1. Graph f and these polynomials on a common screen. (b). Evaluate f and these polynomials at x = 0.9 and 1.3. (c). Comment on how the Taylor polynomials converge to f (x).

> Use series to evaluate the following limit. limx→0 sin x – x/ x3

> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)

> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)

> Use series to approximate f10√1 + x4, dx correct to two decimal places.

> Evaluate f ex/x dx as an infinite series.

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Sketch the curve and find the area that it encloses. r = 2 + cos 2θ

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. x? f

> Find the Taylor series of f (x) = cos x at a = π/3.

> Find the Taylor series of f (x) = sin x at a π/6.

> Find the radius of convergence of the series ∑∞n=1 (2n)!/(n!)2 xn.

> Find the radius of convergence and interval of convergence of the series. 2"(х — 3)" Σ In + 3 00

> Find the radius of convergence and interval of convergence of the series. 2"х — 2)" A-1 (п + 2)!

> Sketch the curve and find the area that it encloses. r = 2 cos 3θ

> Find the radius of convergence and interval of convergence of the series. (x + 2)* Σ n4" 00 n-

> Find the radius of convergence and interval of convergence of the series. x' E (-1)" - n25" ,2.

> Prove that if the series ∑∞n=1 an is absolutely convergent, then the series ∑∞n=1 (n + 1/n)an is also absolutely convergent.

> (a). Show that the series ∑∞n=1nn/(2n)! is convergent. (b). Deduce that limn→∞ nn/(2n)! = 0.

> Use the sum of the first eight terms to approximate the sum of the series∑∞n=1 (2 + 5n)-1. Estimate the error involved in this approximation.

> (a). Find the partial sum s5 of the series ∑∞n=1 1/n6 and estimate the error in using it as an approximation to the sum of the series. (b). Find the sum of this series correct to five decimal places.

> Find the sum of the series ∑∞n=1 (-1)n+1 /n5 correct to four decimal places.

> For what values of does the series ∑∞n=1(ln x)n converge?

> Express the repeating decimal as a 1.2345345345… fraction.

> Find the sum of the series. 1 – e + e2/2! + e3/3! + e4/4! - …

> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 2 <rs 5, 37/4 < 0 < 5T/4

> Find the sum of the series. ∑∞n=1 [tan-1 (n + 1) – tan-1 n]

> Find the sum of the series. ∑∞n=1 (-1)n πn/32n (2n)!

> Find the sum of the series. ∑∞n=1 (-3)n-1/23n

> Determine whether the series is convergent or divergent. ∑∞n=1 (-5)2n/n29n

> Determine whether the series is convergent or divergent. ∑∞n=1 1 ∙ 3 ∙5 ∙ ∙∙∙ (2n – 1)/5nn!

> Determine whether the series is convergent or divergent. ∑∞n=1 cos 3n/1 + (1.2)n

> Determine whether the series is convergent or divergent. ∑∞n=1 (-1)n-1 √n/n + 1

> Determine whether the series is convergent or divergent. ∑∞n=1 ln (n/3m + 1)

> Determine whether the series is convergent or divergent. ∑∞n=1 1/n √ln n

> Determine whether the series is convergent or divergent. ∑∞n=1 (-1)n/√n + 1

> Sketch the curve and find the area that it encloses. r2 = 4 cos 2θ

> Determine whether the series is convergent or divergent. ∑∞n=1 n3/5n

> Determine whether the series is convergent or divergent. ∑∞n=1 n2 +1/n3 + 1

> Determine whether the series is convergent or divergent. ∑∞n=1 n/n3 + 1

> A sequence is defined recursively by the equations a1 = 1, an+1 = 1/3(an + 4). Show that {an} is increasing and an < 2 for all n. Deduce that {an} is convergent and find its limit.

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. {(1 + 3/n)tr}

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. In n an

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. n sin n an ,2 + 1

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. an = = cos(nT/2)

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 3 n a 1+ п? ,2

> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 94+1 an 10"

2.99

See Answer