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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number.f(x) = 1/√x with a = 4; approximate 1/√3.
Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number.f(x) = 3√x with a = 64; approximate 3√60.
Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number.f(x) = √x with a = 36; approximate √39.
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = 1/x4 + 2x2 + 1
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = bx, for b > 0, b ≠ 1
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = √1 - x2
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = x2 cos x2
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = (1 + x2)-2/3
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. sin x if x + 0 f(x) if x = 0 1
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. et + e* -X f(x)
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series.f(x) = cos 2x + 2 sin x
Determine whether the following statements are true and give an explanation or counterexample.a. The function f(x) = √x has a Taylor series centered at 0.b. The function f(x) = csc x has a Taylor series centered at π/2.c. If f has a Taylor series that converges only on (-2, 2), then f(x2) has a
Find the remainder in the Taylor series centered at the point a for the following functions. Then show thatfor all x in the interval of convergence.f(x) = e-x, a = 0 lim R„(x) = 0
Find the remainder in the Taylor series centered at the point a for the following functions. Then show thatfor all x in the interval of convergence.f(x) = cos 2x, a = 0 lim R„(x) = 0
Find the remainder in the Taylor series centered at the point a for the following functions. Then show thatfor all x in the interval of convergence.f(x) = sin x, a = 0 lim R„(x) = 0
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series1/(1 + 4x2)2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series1/(3 + 4x)2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series(x2 - 4x + 5)-2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series1/(1 + 4x2)2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series1/(1- 4 x)2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series(1 + 4x)-2 4x + ·.., for -1 < x < 1. (1 + x)¯² = 1 – 2x + 3x²
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√4 - 16x2 x3 V1 + x = 1 + for –1 < x < 1. 8.
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√a2 + x2, a > 0 x3 V1 + x = 1 + for –1 <
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√1 - 4x x3 V1 + x = 1 + for –1 < x < 1. 8. 16
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√9 - 9x x3 V1 + x = 1 + for –1 < x < 1. 8. 16
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√4 + x x3 V1 + x = 1 + for –1 < x < 1. 8. 16
Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series√1 + x2 x3 V1 + x = 1 + for –1 < x < 1. 8. 16
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = (1 + x)2/3; approximate 1.022/3.
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = (1 + x)-2/3; approximate 1.18-2/3.
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = (1 + x)-3; approximate 1/1.331 = 1/1.13.
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = 4√1 + x; approximate 4√1.12
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = √1 + x; approximate √1.06
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four nonzero terms of the series to approximate the given quantity.f(x) = (1 + x)-2; approximate 1/1.21 = 1/1.12.
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.cosh 3x
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.sinh x2
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.x tan-1 x2
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.(1 + x4)-1
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.cos x3
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. e* – 1 if x # 0 х if x = 0
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.ln (1 + 2x)
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.1/1 - 2x
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.sin x2
Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.ln (1 + x2)
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = 10x, a = 2
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = 2x, a = 1
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = ex, a = ln 2
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = ln x, a = 3
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = 1/x, a = 1
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = cos x, a = π
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = sin x, a = π/2
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = e2x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = sinh 2x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = cosh 3x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = log3(x + 1)
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = 3x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = sin 3x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = tan-1 x/2
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = (1 + 2x)-1
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = ln (1 + 4x)
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = (1 + x2)-1
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = cos 2x
a. Find the first four nonzero terms of the Maclaurin series for the given function.b. Write the power series using summation notation.c. Determine the interval of convergence of the series.f(x) = e-x
Write the Maclaurin series for e2x.
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f ?
For what values of p does the Taylor series for f(x) = (1 + x)p centered at 0 terminate?
Suppose you know the Maclaurin series for f and it converges for |x| < 1. How do you find the Maclaurin series for f(x2) and where does it converge?
How do you find the interval of convergence of a Taylor series?
How do you find the coefficients of the Taylor series for f centered at a?
What conditions must be satisfied by a function f to have a Taylor series centered at a?
How are the Taylor polynomials for a function f centered at a related to the Taylor series for the function f centered at a?
Consider the following function and its power series:a. Let Sn(x) be the sum of the first n terms of the series. With n = 5 and n = 10, graph f(x) and Sn(x) at the sample points x = -0.9, -0.8, . . , -0.1, 0, 0.1, . . , 0.8, 0.9 (two graphs). Where is the difference in the graphs the
Given the power seriesfor -1 < x < 1, find the power series for f(x) = sin-1 x centered at 0. 1.3 x² + 2•4 1•3•5 V1 – x² 2•4•6
Leta. Multiply the power series together as if they were polynomials, collecting all terms that are multiples of 1, x, and x2. Write the first three terms of the product f(x)g(x).b. Find a general expression for the coefficient of xn in the product series, for n = 0, 1, 2, . . . ο0 ο0 and
LetThe remainder in truncating the power series after n terms is Rn(x) = f(x) - Sn(x), which depends on x.a. Show that Rn(x) = xn/(1 - x).b. Graph the remainder function on the interval |x| < 1 for n = 1, 2, 3. Discuss and interpret the graph. Where on the interval is |Rn(x)| largest?
Prove that if converges with radius of convergence R, then the power series for xm f(x) also converges with radius of convergence R, for positive integers m. Σαχό f(x) k=0
We show that the power series for the exponential function centered at 0 isUse the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series.f(x) = x2ex Σ e* = for k! 00 k=0
We show that the power series for the exponential function centered at 0 isUse the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series.f(x) = e-3x Σ e* = for k! 00 k=0
We show that the power series for the exponential function centered at 0 isUse the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series.f(x) = e2x Σ e* = for k! 00 k=0
We show that the power series for the exponential function centered at 0 isUse the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series.f(x) = e-x Σ e* = for k! 00 k=0
Replace x with x - 1 in the series to obtain a power series for ln x centered at x = 1. What is the interval of convergence for the new power series? (-1)*+lx' Σ k+1,k In (1 + x) = k=1 8
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) k 1 3 k=0 8.
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) (x – 2)k 32k k=1
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) Σε ,-kx k=0
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) r2k 4k k=1 8.
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.)
Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) 2k Σα + 1). k=0
If the power series f(x) = ∑ck xk has an interval of convergence of |x| < R, what is the interval of convergence of the power series for f (x - a), where a ≠ 0 is a real number?
If the power series f(x) = ∑ck xk has an interval of convergence of |x| < R, what is the interval of convergence of the power series for f(ax), where a ≠ 0 is a real number?
Write the following power series in summation (sigma) notation. x² .8 x+ х6 1! 2! 3! 4!
Write the following power series in summation (sigma) notation. .7 х х х 4 16
Write the following power series in summation (sigma) notation. 4 3. 2.
Write the following power series in summation (sigma) notation. 3 4 6.
Find the radius of convergence of k!xk kk
Find the radius of convergence of E(1 + k
Determine whether the following statements are true and give an explanation or counterexample.a. The interval of convergence of the power series ∑ck (x - 3)k could be (-2, 8).b. The series ∑(-2x)k converges on the interval -1/2 < x < 1/2.c. If f(x) = ∑ck xk on the interval |x| < 1,
Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.f(x) = tan-1 (4x2)
Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.f(x) = ln √4 - x2
Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.f(x) = ln √1 - x2
Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. 3 f(x) 3 + x
Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x) 1 – x*

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