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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

In Exercises determine the values of for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x) = e 2x 1 - 2x + 2x² - 4 3x³
In Exercises determine the values of for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x) = sin xx x3 3!
Let(a) Find the interval of convergence of ƒ.(b) Show that ƒ'(x) = ƒ(x).(c) Show that ƒ(0) = 1.(d) Identify the function ƒ. f(x) = xn Σ n!' n=0
State the definition of an nth-degree Taylor polynomial of ƒ centered at c.
Match the graph of the first 10 terms of the sequence of partial sums of the serieswith the indicated value of the function. The graphs are labeled (i), (ii), (iii), and (iv). Explain how you made
When an elementary function ƒ is approximated by a second-degree polynomial P₂ centered at c, what is known about ƒ and P₂ at c?Explain your reasoning.
An elementary function is approximated by a polynomial. In your own words, describe what is meant by saying that the polynomial is expanded about c or centered at c.
In Exercises determine the values of for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x) = e = 1 + x + 2! + 3!' x < 0
Write a power series that has the indicated interval of convergence. Explain your reasoning.(a) (-2, 2) (b) (-1, 1](c) (-1, 0)(d) [-2, 6)
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.0001. Use a computer algebra
Give examples that show that the convergence of a power series at an endpoint of its interval of convergence may be either conditional or absolute. Explain your reasoning.
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.0001. Use a computer algebra
Describe how to differentiate and integrate a power series with a radius of convergence R. Will the series resulting from the operations of differentiation and integration have a different radius of
In Exercises find the intervals of convergence of (a) ƒ(x) (b) ƒ'(x)(c) ƒ"(x)(d) ∫ ƒ(x) dx.Include a check for convergence at the endpoints of the interval. f(x) = n=1 (-1)+¹(x -
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of to be less than 0.001.ln (1. 25)
Describe the three basic forms of the domain of a power series.
Describe the interval of convergence of a power series.
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of to be less than 0.001.e0.6
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of to be less than 0.001.cos (0.1)
In Exercises use Taylor’s Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. e 1+1+ 12 13 14 + + 2! 3! 4! + 15 5!
In Exercises use Taylor’s Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. arctan(0.4) 0.4 (0.4) ³ 3
In Exercises use Taylor’s Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. cos(0.3)~ 1 - (0.3)² (0.3)4 + 2! 4!
In Exercises use Taylor’s Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. arcsin (0.4) ≈ 0.4 + (0.4) ³ 2.3
Describe the radius of convergence of a power series.
In Exercises find the intervals of convergence of (a) ƒ(x) (b) ƒ'(x)(c) ƒ"(x)(d) ∫ ƒ(x) dx.Include a check for convergence at the endpoints of the interval. f(x) = ♥ n=0 (−1)n+1(x
In Exercises find the intervals of convergence of (a) ƒ(x) (b) ƒ'(x)(c) ƒ"(x)(d) ∫ ƒ(x) dx.Include a check for convergence at theendpoints of the interval. f(x) = n=0 3 n
In Exercises determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of to be less than 0.001.sin (0. 3)
In Exercises approximate the function at the given value of x, using the polynomial found in the indicated exercise. f(x) = x-cos x,f 이다. ㅇㅇ 8
In Exercises find the intervals of convergence of (a) ƒ(x) (b) ƒ'(x)(c) ƒ"(x)(d) ∫ ƒ(x) dx.Include a check for convergence at the endpoints of the interval. f(x) = n=1 (-1)+¹(x -
Define a power series centered at c.
In Exercises write an equivalent series with the index of summation beginning at n = 1. Ÿ n=0 (-1) x2n+1 2n + 1
In Exercises write an equivalent series with the index of summation beginning at n = 1. n=0 x2n+1 (2n + 1)!
In Exercises approximate the function at the given value of x, using the polynomial found in the indicated exercise. - (³) £ *x_2₂x = (x) £
In Exercises approximate the function at the given value of x, using the polynomial found in the indicated exercise. $(1)- f(x) = ex, fl
In Exercises approximate the function at the given value of x, using the polynomial found in the indicated exercise. f(x) = In x, f(2.1)
In Exercises write an equivalent series with the index of summation beginning at n = 1. 18 0=U n uX n!
In Exercises write an equivalent series with the index of summation beginning at n = 1. 00 n=0 (-1)²+¹(n + 1)x²
In Exercises the graph of y = ƒ(x) is shown with four of its Maclaurin polynomials. Identify the Maclaurin polynomials and use a graphing utility to confirm your results. -2 2 -1 v = In (x2 +
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 k(k + 1)(k + 2) · · · (k + n − 1)xn n! k≥ 1
In Exercises the graph of y = ƒ(x) is shown with four of its Maclaurin polynomials. Identify the Maclaurin polynomials and use a graphing utility to confirm your results. 2 y =
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 n!(x - c)n 1.3.5 (2n - 1) .
In Exercises the graph of y = ƒ(x) is shown with four of its Maclaurinpolynomials. Identify the Maclaurin polynomials and use agraphing utility to confirm your results. -6 6- 4 2 -4 -6+ y = cos
In Exercises (a) Find the Maclaurin polynomial P3(x) for ƒ(x) (b) Complete the table for ƒ(x) and P3(x)(c) Sketch the graphs of ƒ(x) and P3(x) on the same set of coordinate axes.ƒ(x) =
In Exercises the graph of y = ƒ(x) is shown with four of its Maclaurin polynomials. Identify the Maclaurin polynomials and use a graphing utility to confirm your results. +3 -3-2 2 1 y -2. y =
In Exercises find the radius of convergence of the power series, where c > 0 and k is a positive integer. n=0 (n!)k xn (kn)!
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (-1)+¹(xc)₂ nch
In Exercises find the radius of convergence of the power series, where c > 0 and k is a positive integer. Σ n=1 (x – c)n-1 Ch-1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 n! (x + 1)^ 1.3.5 (2n - 1)
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 2.4.6. 2n 3.5.7 (2n + 1). . ▼ x2n+1
In Exercises (a) Find the Maclaurin polynomial P3(x) for ƒ(x) (b) Complete the table for ƒ(x) and P3(x)(c) Sketch the graphs of ƒ(x) and P3(x) on the same set of coordinate axes.ƒ(x) =
In Exercises use a computer algebra system to find the indicated Taylor polynomials for the function ƒ. Graph the function and the Taylor polynomials.(a) n = 4, c = 0(b) n = 4, c = 1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 2 3 4 (n + 1)xn n!
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (-1)+137 11 (4n 1)(x − 3) 4n -
In Exercises find the nth Taylor polynomial centered at c. f(x) = x² cos x, n = 2, c = π
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 n!xn (2n)!
In Exercises find the nth Taylor polynomial centered at c. f(x) = 3√x, n = 3, c = 8
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 x³n+1 (3n+ 1)!
In Exercises find the nth Taylor polynomial centered at c. f(x) = ln x, n = 4, c = 2
In Exercises find the nth Taylor polynomial centered at c. f(x)=√x, n = 3, c = 4
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (-1)n x²n n!
In Exercises use a computer algebra system to find the indicated Taylor polynomials for the function ƒ. Graph thefunction and the Taylor polynomials.ƒ(x) = tan πx(a) n = 3, c = 0(b) n = 3, c
In Exercises find the nth Taylor polynomial centered at c. f(x) = x²² n = 4, c = 2
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 n n+ 1 -(-2x)n-1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (x - 3)-1 3n-1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (-1) x²n+1 2n + 1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (-1)+1(x - 1)n+1 n+ 1
In Exercises find the nth Taylor polynomial centered at c. 2 f(x) == x² n = 3, c = 1
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = sec x, n = 2
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (-1)+¹(x - 2)" n2"
In Exercises find the nth Maclaurin polynomial for the function. f(x) = X x + 1' n = 4
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (x − 3)+1 (n + 1)4n+1
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) Σ n=0 (−1)" n!(x − 5)n 3″
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = tan x, n = 3
In Exercises find the nth Maclaurin polynomial for the function. f(x) = 1 x + 1' n = 5
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (-1)+1xn 6⁰
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (-1) xn (n + 1)(n + 2)
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (3x)n (2n)!
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 Sn n!
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) Σ n = 0 (2n)! 3, Π
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = x²e-x, n = 4
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 (−1)n+¹(n+1)xn
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = xex, n = 4
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1 (-1) xn n
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = cos πX, n = 4
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = sin x, n = 5
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0 X 4, n
In Exercises use a graphing utility to graph ƒ and its second-degree polynomial approximation P₂ at x = c. Complete the table comparing the values of ƒ and P₂. f(x) = = sec
In Exercises find the radius of convergence of the power series. n=0 (2n)!x2n n!
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = ex/³, n = 4
In Exercises find the radius of convergence of the power series. n=0 X2п (2n)!
In Exercises find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) (2x)n n=0
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = e-x/2, n = 4
In Exercises use a graphing utility to graph ƒ and its second-degreepolynomial approximation P₂ at x = c. Complete the tablecomparing the values of ƒ and P₂. f(x) = 4 X x P₂(x) = 42(x - 1) +
In Exercises find the radius of convergence of the power series. n=0 (-1) xn 5n
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = e-x, n = 5
In Exercises find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of ƒ at x = c. Use a graphing utility to graph ƒ and P₁. What is P₁ called?
In Exercises find the radius of convergence of the power series. n=1 (4x)n n²
In Exercises find the nth Maclaurin polynomial for the function.ƒ(x) = e4x, n = 4
In Exercises find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of ƒ at x = c. Use a graphing utility to graph ƒ and P₁. What is P₁ called?
In Exercises find a first-degree polynomial function P₁ whose value and slope agree with the value and slope of ƒ at x = c. Use a graphing utility to graph ƒ and P₁. What is P₁ called?

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