15. Power Series

{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

{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

{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

{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

{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

{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

Approximating ln 2 Consider the following three ways to approximate

ln 2.

c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.

Approximating ln 2 Consider the following three ways to approximate

ln 2.

b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.

Approximating ln 2 Consider the following three ways to approximate

ln 2.

a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.

Approximating ln 2 Consider the following three ways to approximate

ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

Approximating ln 2 Consider the following three ways to approximate

ln 2.

d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

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).

{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

b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

{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

c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

{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

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?