91Ó°ÊÓ

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. $$y^{\prime}(t)-3 y=10, y(0)=2$$

a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\ln x, a=e$$

Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0 . $$\left(1+x^{4}\right)^{-1}$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(3 x)=\ln (1-3 x)$$

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. $$y^{\prime}(t)=6 y+9, y(0)=2$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$g(x)=x^{3} \ln (1-x)$$

a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt[4]{x}, a=16$$

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}.\) $$\int_{0}^{0.25} e^{-x^{2}} d x$$

a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\tan ^{-1} x+x^{2}+1, a=1$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$h(x)=x \ln (1-x)$$