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08:44L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.
a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.
b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.
{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]
Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is
eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞
Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.
f(x) = e⁻³ˣ
Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.
f(x) = sin x, a = π/2
{Use of Tech} Approximating powers 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=64; approximate ∛60.
Inverse sine Given the power series
1/√(1 − x²) = 1 + (1/2)x² + (1 ⋅ 3)/(2 ⋅ 4) x⁴ + (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) x⁶ +⋯
for −1<x<1, find the power series for f(x) = sin ⁻¹ x centered at 0.
