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Problem 11.3.15a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = (1 + x²)⁻¹, a = 0
Problem 11.3.19a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = tan ⁻¹ (x/2), a = 0
Problem 11.2.63a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).
Problem 11.1.67a
Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.
a. √(1 + 2x)
A. p₂(x)= 1 + 2x + 2x²
B. p₂(x) = 1 − 6x + 24x²
C. p₂(x) = 1 + x − x²/2
D. p₂(x) = 1 − 2x + 4x²
E. p₂(x) = 1 − x + (3/2)x²
F. p₂(x) = 1 − 2x + 2x²
Problem 11.3.67a
Explain why or why not 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.
Problem 11.3.29a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = 1/x, a = 1
Problem 11.3.13a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=2/(1−x)³, a=0
Problem 11.3.33a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = 2ˣ, a = 1
Problem 11.3.11a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = e⁻ˣ, a=0
Problem 11.3.17a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = e²ˣ, a = 0
Problem 11.1.75a
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = eˣ ≈ 1 + x
Problem 11.3.23a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = cosh 3x, a = 0
Problem 11.4.79a
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
a. Compute S′(x) and C′(x).
Problem 11.3.25a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = ln (x − 2), a = 3
Problem 11.4.76a
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.
Problem 11.3.27a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=sin x, a = π/2
Problem 11.1.65a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.
Problem 11.3.31a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = ln x, a = 3
Problem 11.4.78a
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.
Problem 11.4.81a
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
a. Write out the first four terms of J₀.
Problem 11.1.76b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin ⁻¹ x ≈ x
Problem 11.3.49b
{Use of Tech} Binomial series
b. Use the first four terms of the series to approximate the given quantity.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
Problem 11.3.21b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=3ˣ, a=0
Problem 11.4.81b
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
b. Find the radius and interval of convergence of the power series for J₀.
Problem 11.1.74b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = ln (1 + x) ≈ x − x²/2
Problem 11.1.70b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = tan x ≈ x
Problem 11.1.69b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin x ≈ x
Problem 11.3.9b
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to 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/x², a=1
Problem 11.3.67b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.
Problem 11.4.76b
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.