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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = √x, a = 9
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a. f(x) = tan x, a = 74
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = tan x, a = 0
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = e−x + e−2x, a = 0
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = e2x, a = ln 2
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = x2e−x, a = 1
First calculate and evaluate the needed derivatives: f(x) = Inx X a = 1
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = cosh 2x, a = 0
First calculate and evaluate the needed derivatives:ƒ(x) = ln(x + 1), a = 0
Show that the second Taylor polynomial for ƒ(x) = px2 + qx + r, centered at a = 1, is ƒ(x).
Show that the nth Maclaurin polynomial for ƒ(x) = ex is X Tn(x)=1+ + 1! 2! n!
Show that the third Maclaurin polynomial for ƒ (x) = (x − 3)3 is ƒ(x).
Show that the nth Taylor polynomial for ƒ(x) = 1/x + 1 at a = 1 is (x-1) Ta(x) = 12 - (x - 1)+ (x-13² 4 8 + · + (−1)n (x − 1)n 2n+1
Show that the Maclaurin polynomials for ƒ(x) = sin x are T2n+1(x) = T2n+2(x) = x- - x³ 3! + 5! 2n+1 (2n + 1)! + (−1)"-
Show that the Maclaurin polynomials for ƒ(x) = ln(1 + x) are Tn(x) = x +² 2 + 3 + ·· + (-1)^- 1x² n
Find Tn centered at x = a for all n. f(x) = 1 1 + x a=0
Find Tn centered at x = a for all n. f(x) = 1 x - l' a = 4
Find Tn centered at x = a for all n.ƒ(x) = ex, a = 1
Find Tn centered at x = a for all n.ƒ(x) = ex, a = −2
Find Tn centered at x = a for all n.ƒ(x) = x−2, a = 1
Find Tn centered at x = a for all n. f(x) = cos x, a = π 4
Find Tn centered at x = a for all n.ƒ(x) = x−2, a = 2
Find Tn centered at x = a for all n.ƒ(θ) = sin 3θ, a = 0
Find T2 and use a calculator to compute the error |ƒ(x) − T2(x)| for the given values of a and x. = 0, a = 0, y = cos x, a x= π 12
Find T2 and use a calculator to compute the error |ƒ(x) − T2(x)| for the given values of a and x.y = ex, a = 0, x = −0.5
Find T2 and use a calculator to compute the error |ƒ(x) − T2(x)| for the given values of a and x. y = esin.x a = KIN 2² x = 1.5
Find T2 and use a calculator to compute the error |ƒ(x) − T2(x)| for the given values of a and x.y = x−2/3, a = 1, x = 1.2
Compute T3 for ƒ(x) = √x centered at a = 1. Then use a plot of the error |ƒ(x) − T3(x)| to find a value c > 1 such that the error on the interval [1, c] is at most 0.25.
Plot ƒ(x) = 1/(1 + x) together with the Taylor polynomials Tn at a = 1 for 1 ≤ n ≤ 4 on the interval [−2, 8] (be sure to limit the upper plot range).(a) Over which interval does T4 appear to approximate ƒ closely?(b) What happens for x < −1?(c) Use your computer algebra system to produce
Let T3 be the Maclaurin polynomial of ƒ(x) = ex. Use the Error Bound to find the maximum possible value of |ƒ (1.1) − T3(1.1)|. Show that we can take K = e1.1.
Use the Ratio Test to determine the radius of convergence Does it converge at the endpoints x = ±R? 00 Ror Σ n=0 the 2n
Let T2 be the Taylor polynomial of ƒ(x) = √x at a = 4. Apply the Error Bound to find the maximum possible value of the error |ƒ(3.9) − T2(3.9)|.
Repeat Exercise 3 for the following series: (a) n=1 (x (x - 5)" 9n (b) n=1 (x - 5)" n9n (c) 00 n=1 (x - 5)" n²9n
Show thatdiverges for all x ≠ 0. 00 Σπ.χ n. n = 0 Η
For which values of x does 00 n=0 n!x" converge?
Use the Ratio Test to show thathas radius of convergence R =√3. 00 n=0 x2n X 3n
Show that has radius of convergence R = 4. Σ n=0 3n+1 64n
Find the interval of convergence. 00 Σηχ" nx n=0
Find the interval of convergence. 00 n=1 2n th n
Find the interval of convergence. 00 Σ+1) n=1 x2n+1 2"n
Find the interval of convergence. Σ+1" η η 2η n 4n n=0
Find the interval of convergence. 00 Σ n=4 εγω της
Find the interval of convergence. 00 n=8 7 n'x
Find the interval of convergence. 00 Σ n=0 xh (n!)²
Find the interval of convergence. 00 Σ n=0 gn n! th
Find the interval of convergence. 00 n=0 (2n)! n (n!)3
Find the interval of convergence. 00 (−1)"x" Σ n=o Vn2 + 1
Find the interval of convergence. 00 Σ n=0 xn nt + 2
Find the interval of convergence. 00 Σ n=15 x2n+1 3n + 1
Find the interval of convergence. 00 Σ; h=9 x" n - 4 ln n
Find the interval of convergence. 00 Σ n=2 χ In n
Find the interval of convergence. 00 n=2 x3n+2 In n
Find the interval of convergence. 00 Σna – 3)" n(x n=1
Find the interval of convergence. 8 (−5)" (x – 3)" Σ n2 n=1
Find the interval of convergence. 00 Σ n=0 (x - 4) n!
Find the interval of convergence. 00 Σ -(x + 10) (-5)" n! n=0
Find the interval of convergence. 00 En!(x+5) n! (x + 5)" n=10
Find the interval of convergence. 00 Σ n=2 (x + 4) (n Inn)2
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
(a) Divide the power series in Exercise 42 by 4x3 to obtain a power series for h(x) = 1/(1 − x4)2 and use the Ratio Test to show that the radius of convergence is 1.Data From Exercise 42Differentiate the power series in Exercise 40 to obtain a power series for Data From Exercise 40Use Eq. (2)
Show that the following series converges conditionally: 1 n2/3 Σ+19-1. H=1 = 1 12/3 1 22/3 + 1 32/3 1 42/3 +
Determine whether the series converges absolutely, conditionally, or not at all. 8 n=1 (−1)n4 n³ + 1
Determine whether the series converges absolutely, conditionally, or not at all. 8W Σ n=0 (-1)" (1.001)n
Determine whether the series converges absolutely, conditionally, or not at all. n=1 (-1)"e-n n²
Determine whether the series converges absolutely, conditionally, or not at all. 8 n=1 sin() n²
Determine whether the series converges absolutely, conditionally, or not at all. n=2 (-1)" n ln n
Determine whether the series converges absolutely, conditionally, or not at all. 8 n=1 (-1)" 1 + 1/1 n
Let (a) Calculate Sn for 1 ≤ n ≤ 10.(b) Use the inequality in (2) to show that 0.9 ≤ S ≤ 0.902. S = [(-1)"+¹_ n=1 n
Use the inequality in (2) to approximateto four decimal places. S-SN|
Approximate to three decimal places. 00 n=1 (−1)n+1
Let Use a computer algebra system to calculate and plot the partial sums S n for 1 ≤ n ≤ 100. Observe that the partial sums zigzag above and below the limit. 00 n S = [(-1)-1_ n² + 1 n=1
Find a value of N such that SN approximates the series with an error of at most 10−5. Using technology, compute this value of SN. 8 (-1)+1 | n(n + 2)(n + 3) Σ n=1
Determine convergence or divergence by any method. 00 Στη n=0
Determine convergence or divergence by any method. 00 n=1 1 27.5
Determine convergence or divergence by any method. n=1 1 5n3n
Determine convergence or divergence by any method. 00 n=1 1 n+ n
Determine convergence or divergence by any method. 00 n=0 (-1)^n √n² + 1
Determine convergence or divergence by any method. n=1 (−1)"+1 (2n + 1)!
Determine convergence or divergence by any method. Σ+19ne-3/3 n=1
Determine convergence or divergence by any method. 00 n=1 he 13/3 ne
Determine convergence or divergence by any method. 8 n=2 (-1)" n¹/2(Inn)²
Determine convergence or divergence by any method. 00 1 Σ n(Inn)1/4 n=2
Determine convergence or divergence by any method. n=1 In n 21.05 n
Determine convergence or divergence by any method. 00 n=2 1 (Inn)² 2
Show thatconverges by computing the partial sums. Does it converge absolutely? 1 - IN 2 1 2 1 + · Im 3 | 113 + 1 4 1 4
The Alternating Series Test cannot be applied toWhy not? Show that it converges by another method. 1 2 1 1 + 3 2² 1 3² 1 23 1 33
Determine whether the following series converges conditionally: 1 3 + 1 2 T 1 + 5 3 7 + 1 4 1 1 + 9 5 667 11 +
Use the Integral Test to determine whether the infinite series is convergent. 1 Σ (n + 1)4 n=1
Prove that if ∑an converges absolutely, then ∑a2n also converges. Give an example where ∑an is only conditionally convergent and ∑a2n diverges.
Use the Integral Test to determine whether the infinite series is convergent. Μ8 Σ h=1 n η (n2 + 1)3/5
Use the Integral Test to determine whether the infinite series is convergent. 00 n=1 1 n² + 1
Use the Integral Test to determine whether the infinite series is convergent. Σ H=4 1 n2 – 1 -
Use the Integral Test to determine whether the infinite series is convergent. (S+u)u I 1=u 3 00
Use the Integral Test to determine whether the infinite series is convergent. 00 Σ H=1 ne
Use the Integral Test to determine whether the infinite series is convergent. Σ n=2 1 n(Inn)3/2
Use the Integral Test to determine whether the infinite series is convergent. 00 n=1 Inn n²
Forverify that for n ≥ 1, 00 n=1 1 n+ √n'

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