91Ó°ÊÓ

How are the Taylor polynomials for a function \(f\) centered at \(a\) related to the Taylor series for the function \(f\) centered at \(a ?\)

Write the first four terms of a power series with coefficients \(c_{0}, c_{1}\) \(c_{2},\) and \(c_{3}\) centered at 0

How do you find the coefficients of the Taylor series for \(f\) centered at \(a ?\)

Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.

In terms of the remainder, what does it mean for a Taylor series for a function \(f\) to converge to \(f ?\)

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x}$$

a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. $$f(x)=\cos x, a=\pi / 4 ; \text { approximate } \cos (0.24 \pi)$$

Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. $$y^{\prime}(t)-y=0, y(0)=2$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.$$\tan (-0.1)$$ c.\(T_{3}(z)=0.33333333 x^{3}+x\)