91Ó°ÊÓ

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

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

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

the Taylor polynomial \(P_{n}(x)\) about 0 approximates \(f(x)\) with error \(E_{n}(x)\) and the Taylor series converges to \(f(x) .\) Find the smallest constant \(K\) given by the alternating series error bound such that \(\left|E_{4}(1)\right| \leq K\) $$f(x)=\cos x$$

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

For Exercises \(10-17\), find the first four terms of the Taylor series for the function about the point \(a\). $$\sin x, \quad a=\pi / 4$$

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.) $$(1+x)^{p}, \quad n=2,3,4$$

the Taylor polynomial \(P_{n}(x)\) about 0 approximates \(f(x)\) with error \(E_{n}(x)\) and the Taylor series converges to \(f(x) .\) Find the smallest constant \(K\) given by the alternating series error bound such that \(\left|E_{4}(1)\right| \leq K\) $$f(x)=\sin x$$

Find the constant term of the Fourier series of the triangular wave function defined by \(f(x)=|x|\) for \(-1 \leq\) \(x \leq 1\) and \(f(x+2)=f(x)\) for all \(x\).