91Ó°ÊÓ

Approximations with Taylor polynomials a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator. Approximate \(e^{-0.15}\) using \(f(x)=e^{-x}\) and \(p_{2}(x)=1-x+x^{2} / 2\).

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. b. Write the power series using summation notation. $$f(x)=\ln x, a=3$$

Power series for derivatives a. Differentiate the Taylor series centered at 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=\tan ^{-1} x$$

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. b. Write the power series using summation notation. $$f(x)=e^{x}, a=\ln 2$$

Approximations with Taylor polynomials a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator. Approximate \(\ln 1.06\) using \(f(x)=\ln (1+x)\) and \(p_{2}(x)=x-x^{2} / 2\).

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty}\left(-\frac{x}{10}\right)^{2 k}$$

Power series for derivatives a. Differentiate the Taylor series centered at 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=-\ln (1-x)$$

Approximations with Taylor polynomials 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. $$e^{0.12}$$

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. b. Write the power series using summation notation. $$f(x)=2^{x}, a=1$$

Differential equations at 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. $$y^{\prime}(t)-y=0, y(0)=2$$