91Ó°ÊÓ

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\phi^{3} \cos \left(\phi^{2}\right)$$

(a) Using a calculator, make a table of values to four decimal places of \(\sin x\) for $$x=-0.5,-0.4, \ldots,-0.1,0,0.1, \ldots, 0.4,0.5$$ (b) Add to your table the values of the error \(E_{1}=\) \(\sin x-x\) for these \(x\) -values. (c) Using a calculator or computer, draw a graph of the quantity \(E_{1}=\sin x-x\) showing that $$\left|E_{1}\right|<0.03 \text { for }-0.5 \leq x \leq 0.5$$

Find the first four terms of the Taylor series for the function about the point \(a\). $$\cos \theta, \quad a=\pi / 4$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\frac{1}{\sqrt{1+x}}, \quad n=2,3,4$$

Find the \(n^{\text {th }}\). Fourier polynomial for the given functions, assuming them to be periodic with period \(2 \pi\) Graph the first three approximations with the original function. $$h(x)=\left\\{\begin{array}{ll}0 & -\pi

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\arctan \left(r^{2}\right)$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$(1+x)^{p}, \quad n=2,3,4$$

Find the first four terms of the Taylor series for the function about the point \(a\). $$\cos t, \quad a=\pi / 6$$

Find the \(n^{\text {th }}\). Fourier polynomial for the given functions, assuming them to be periodic with period \(2 \pi\) Graph the first three approximations with the original function. $$g(x)=x, \quad-\pi

Find the first four terms of the Taylor series for the function about the point \(a\). $$\sin \theta, \quad a=-\pi / 4$$