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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = Cos x³/2
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = =(ex – ex) = sinh x -
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = cos 4x
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. g(x) = 2 sin x³
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = COS TTX
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = sin 7x
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. g(x) = sin 3x
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = ln(1 + x)
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises explain how to use the geometric seriesto find the series for the function. Do not find the series. g(x) = 1 1- x =x", |x| < 1 n=0 =
In Exercises find a geometric power series for the function, centered at 0(a) By the technique shown in Examples 1 (b) By long divisionData from in Examples 1 Long Division 2 x + x²x²³+. 2 + x) 4 4 + 2x -2x -2x - x² x² x² + x³² đ EXAMPLE 1 Finding a Geometric Power Series Centered at
In Exercises find a geometric power series for the function, centered at 0(a) By the technique shown in Examples 1 (b) By long divisionData from in Examples 1 Long Division 2 x + x²x²³+. 2 + x) 4 4 + 2x -2x -2x - x² x² x² + x³² đ EXAMPLE 1 Finding a Geometric Power Series Centered at
In Exercises find a geometric power series for the function, centered at 0(a) By the technique shown in Examples 1 (b) By long divisionData from in Examples 1 Long Division 2 x + x²x²³+. 2 + x) 4 4 + 2x -2x -2x - x² x² x² + x³² đ EXAMPLE 1 Finding a Geometric Power Series Centered at
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) 2 6 - x' C = -2
In Exercises find a geometric power series for the function, centered at 0(a) By the technique shown in Examples 1 (b) By long divisionData from in Examples 1 Long Division 2 x + x²x²³+. 2 + x) 4 4 + 2x -2x -2x - x² x² x² + x³² đ EXAMPLE 1 Finding a Geometric Power Series Centered at
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) 1 1 - 3x² c=0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) = 1 3 - x' c = 1
In Exercises find a power series for the function, centered at c and determine the interval of convergence. g(x) = 5 2x - 3' C = -3
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) 3 3x + 4' c=0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. h(x) = 1 1 – 5x C = 0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) = 4 3x + 2' c = 3
In Exercises find a power series for the function, centered at c and determine the interval of convergence. g(x) 4x x² + 2x - 3² c=0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) 3 2x - 1' c=2
In Exercises find a power series for the function, centered at c and determine the interval of convergence. g(x) = 3x - 8 3x² + 5x - 2' c = 0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x) 5 5 + x²¹ c=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises find a power series for the function, centered at c and determine the interval of convergence. f(x): 2 1-x²¹ c=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises use the power seriesto determine a power series, centered at 0, for the function. Identify the interval of convergence. 1 1 + x = = Ž (-1)".x" n=0
In Exercises (a) Graphseveral partial sums of the series(b) Find the sum of the seriesand its radius of convergence(c) Use 50 terms of the series toapproximate the sum when x = 0.5(d) Determine whatthe approximation represents and how good the approximation is n=1 (−1)n (-1)+¹(x - 1)n n
LetUse a graphing utility to confirm the inequality graphically.Then complete the table to confirm the inequality numerically. Sn = x - x² 2 + x3 3 4 + n
LetUse a graphing utility to confirm the inequality graphically.Then complete the table to confirm the inequality numerically. Sn = x - x² 2 + x3 3 4 + n
In Exercises use the series for ƒ(x) = arctan x to approximate the value, using RN ≤ 0.001. 1 arctan 4
In Exercises use the series for ƒ(x) = arctan x to approximate the value, using RN ≤ 0.001. 3/4 0 arctan x2 dx
In Exercises use the series for ƒ(x) = arctan x to approximate the value, using RN ≤ 0.001. 1/2 arctan x2 X - dx
In Exercises (a) Graph several partial sums of the series(b) Find the sum of the series and its radius of convergence(c) Use 50 terms of the series to approximate the sum when x = 0.5(d) Determine what the approximation represents and how good the approximation is ☎ (−1)″ x²n+1 (2n +
In Exercises use the power seriesFind the series representation of the function and determine its interval of convergence. 1 1- x = ·Î x² x", |x| < 1. n=0
In Exercises use the power seriesFind the series representation of the function and determine its interval of convergence. 1 1- x = ·Î x² x", |x| < 1. n=0
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 k n k> 0
In Exercises use the series for ƒ(x) = arctan x to approximate the value, using RN ≤ 0.001. (1/2 x² arctan x dx
In Exercises use the power seriesFind the series representation of the function and determine its interval of convergence. 1 1- x = ·Î x² x", |x| < 1. n=0
In Exercises use the power seriesFind the series representation of the function and determine its interval of convergence. 1 1- x = ·Î x² x", |x| < 1. n=0
In Exercises(a) Verify the given equation(b) Use the equation and the series for the arctangent to approximate π to two-decimal-place accuracy. arctan 1 + arctan 1 3 ㅠ 4
Prove thatfor xy ≠ 1 provided the value of the left side of the equation isbetween -π/2 and π/2. arctan x + arctan y = arctan x + y 1 - xy
Use the result of Exercise 45 to verify each identity.(a)(b)Data from in Exercise 45Prove thatfor xy ≠ 1 provided the value of the left side of the equation is between -π/2 and π/2. arctan 120 119 - arctan 1 239 ㅠ 4
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. -| Σ (-1)^+1. n=1 1 2"n
In Exercises explain how to use the geometric seriesto find the series for the function. Do not find the series. g(x) = 1 1- x =x", |x| < 1 n=0 =
In Exercises explain how to use the geometric seriesto find the series for the function. Do not find the series. g(x) = 1 1- x =x", |x| < 1 n=0 =
In Exercises explain how to use the geometric seriesto find the series for the function. Do not find the series. g(x) = 1 1- x =x", |x| < 1 n=0 =
In Exercises(a) Verify the given equation(b) Use the equation and the series for the arctangent to approximate π to two-decimal-place accuracy. 2 arctan 1 arctan 1 ㅠ 7 4
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. Σ(-1)+1, 1 3"n n=1
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. n=1 2n (−1)n+1- 5″ n
One of the series in Exercises 49 converges to its sum at a much lower rate than the other five series. Which is it? Explain why this series converges so slowly. Use a graphing utility to illustrate the rate of convergence.Data from in Exercises 49In Exercises find the sum of the convergent series
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. n=0 1 2n + 1 (-1)"-
The power seriesconverges forWhat can you conclude about the seriesExplain. 18 n=0 'n -n
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. n=0 1 22n+1(2n + 1) (−1)n-
In Exercises find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. 1 32n-1(2n-1) Σ (-1)+1. n=1
The radius of convergence of the power seriesWhat is the radius of convergence of the seriesExplain. Σax" is 3. n=0
The graphs show first-, second-, and third-degree polynomial approximations P₁, P2, and P3 of a function ƒ. Label the graphs of P₁, P₂, and P3. 3 2 1 y 1 2 3 نرا f X
In Exercises find the sum of the series. (-1) 오 3"(2n + 1) n=0
In Exercises find the sum of the series. n=0 (-1)" 77²n+1 32n+1(2n + 1)!
Use a graphing utility to show that 8 9801 n=0 (4n)!(1103 +26,390n) (n!)3964n TT
Describe why the statement is incorrect. n=0 + ux n=0 X n=0 1 1 + xn 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)+¹(x-4)n n9n
For n > 0, let R > 0 and cn > 0. Prove that if the interval of convergence of the seriesthen the series converges conditionally at n=0 c₂(x - x)" is [xo - R, xo + R], Xo - R. Хо
Prove that if the power serieshas a radius ofconvergence of R, thenhas a radius of convergence of √R. 00 n=0 Cnxn
Letwhere the coefficients are C2n = 1 and C2n + 1 = 2 for n ≥ 0.(a) Find the interval of convergence of the series.(b) Find an explicit formula for g(x). g(x) = 1 + 2x + x² + 2x³ + x4 +
Prove that the power serieshas a radius of convergence of R = ∞ when p and q are positive integers. n=0 (n + p)! n!(n + q)!" 9)! Xn
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Ifconverges for |x| < 2, then f(x) = Σ Σ n=0 anth
Letwhere(a) Find the interval of convergence of the series.(b) Find an explicit formula for ƒ(x). f(x) = Σ cx", n=0
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the interval of convergence foris (-1, 1) then theinterval of convergence for Σ 0 = " n = 0 uX"D
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the power series converges for x = 2, then it also converges for x = -2. Σ n=1 uX D Π
In Exercises the series represents a well-known function. Use a computer algebra system to graph the partial sum S10 and identify the function from the graph. x2n+1 (2n + 1)! f(x) = Σ (1) . – n=0
In Exercises the series represents a well-known function. Use a computer algebra system to graph the partial sum S10 and identify the function from the graph. x2n+1 2n + 1' f(x) = Σ (1)", n=0 - 1 ≤ x ≤ 1
In Exercises the series represents a well-known function. Use a computer algebra system to graph the partial sum S10 and identify the function from the graph. f(x) = Î (−1)″ x", −1
In Exercises the series represents a well-known function. Use a computer algebra system to graph the partial sum S10 and identify the function from the graph. f(x) = n=0 x2n (-1). (2n)!
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.It is possible to find a power series whose interval of convergence is [0, ∞].
The proof in Exercise 70 guarantees that the Taylor polynomial and its derivatives agree with the function and its derivatives at x = c. Use the graphs and tables in Exercises to discuss what happens to the accuracy of the Taylor polynomial as you move away from x = c.Data from in Exercise 70Let
In Exercises show that the function represented by the power series is a solution of the differential equation. y = 1 + (-1)n x4n 22n n! 37. 11. · · (4n − 1)' .. n=1 y" + x²y = 0
In Exercises show that the function represented by the power series is a solution of the differential equation. y = n=0 x2n 2" n!' y"-xy' - y = 0
Let Pn(x) be the nth Taylor polynomial for ƒ at c.Prove that Pn (c) = ƒ(c) and P(k)(c) = ƒ(k) (c) for 1 ≤ k ≤n.
Prove that if ƒ is an even function, then its nthMaclaurin polynomial contains only terms with even powersof x.
In Exercises show that the function represented by the power series is a solution of the differential equation. y = n=0 x2n+1 (2n + 1)!' y" - y = 0
Prove that if ƒ is an odd function, then its nthMaclaurin polynomial contains only terms with odd powers of x.
In Exercises show that the function represented by the power series is a solution of the differential equation. y = n=0 x2n (2n)!' y" - y = 0
The graphs show first-, second-, and third-degree polynomial approximations P₁, P2, and P3 of a function ƒ. Label the graphs of P₁, P2, and P3.
In general, how does the accuracy of a Taylor polynomial change as the degree of the polynomial increases? Explain your reasoning.
In Exercises show that the function represented by the power series is a solution of the differential equation. y = Ÿ n=0 (−1)n x²n+1 (2n + 1)!' y" + y = 0
In Exercises show that the function represented by the power series is a solution of the differential equation. y = n=0 (-1)¹ x2n (2n)! y" + y = 0
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) = cos x 1 2! + x4 4!
Describe the accuracy of the nth-degree Taylor polynomial off centered at c as the distance between c and x increases.
Letand(a) Find the intervals of convergence of ƒ and g.(b) Show that ƒ'(x) = g(x).(c) Show that g'(x) = -ƒ(x).(d) Identify the functions ƒ and g. f(x) = Σ n=0 (−1)n x2n + 1 (2n + 1)!
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!
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