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Problem 11.1.56
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
tan x ≈ x on [−π/6, π/6]
Problem 11.1.58
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
√(1+x) ≈ 1 + x/2 on [−0.1,0.1]
Problem 11.1.62
{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85
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 11.1.70b
{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 11.1.71b
{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) = tan⁻¹ x ≈ x
Problem 11.1.73b
{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) =√(1+x) ≈ 1 + x/2
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 11.1.75a
{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 11.1.76b
{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 11.1.83
Tangent line is p₁ Let f be differentiable at x=a
a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).
b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).
Problem 11.1.80
{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?
Problem 11.1.85
{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.
a. Find p₅ and q₅
b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?
c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.
d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.
Problem 11.1.88
{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].
a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.
b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:
q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)
Show that q(x) = −(4/π²)x² + (4/π)x.
c. Graph f, p₂, and q on [0, π].
d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.
e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.
Problem 11.2.57
Functions to power series 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) = 2x/(1 + x²)²
Problem 11.2.61
Functions to power series 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)
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 11.2.60
Functions to power series 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)
Problem 11.2.8
Suppose the power series ∑ₖ₌₀∞ cₖ(x−a)ᵏ has an interval of convergence of (−3,7]. Find the center a and the radius of convergence R.
Problem 11.2.6
Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.
Problem 11.2.2
Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.
Problem 11.2.30
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ k(x−1)ᵏ
Problem 11.2.35
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!
Problem 11.2.39
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = tan⁻¹(4x), a = 0
Problem 11.2.37
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)
Problem 11.2.32
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ
Problem 11.2.40
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ
Problem 11.2.9
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ
Problem 11.2.11
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (kx)ᵏ
Problem 11.2.13
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ sinᵏ(1/k) xᵏ
