Chapter 10

Which of the series are Fourier series? $$1+\cos x+\cos ^{2} x+\cos ^{3} x+\cos ^{4} x+\cdots$$

Use Theorem 10.1 to find a bound for the error in approximating the quantity with a third-degree Taylor polynomial for the given function \(f(x)\) about \(x=0 .\) Compare the bound with the actual error. $$e^{0.1}, f(x)=e^{x}$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$e^{-x}$$

Find the first four nonzero terms of the Taylor series for the function about 0. $$(1+x)^{3 / 2}$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\frac{1}{1-x}, \quad n=3,5,7$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\sqrt{1-2 x}$$

Find the first four nonzero terms of the Taylor series for the function about 0. $$\sqrt[4]{x+1}$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\frac{1}{1+x}, \quad n=4,6,8$$

Which of the series are Fourier series? $$\begin{aligned} &\frac{\cos x}{2}+\sin x-\frac{\cos (2 x)}{4}-\frac{\sin (2 x)}{2}+\frac{\cos (3 x)}{8}+\\\&\frac{\sin (3 x)}{3}-\cdots \end{aligned}$$

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

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

Find the first four nonzero terms of the Taylor series for the function about 0. $$\sin (-x)$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\ln (1-2 y)$$

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

Find the first four nonzero terms of the Taylor series for the function about 0. $$\ln (1-x)$$

Use Theorem 10.1 to find a bound for the error in approximating the quantity with a third-degree Taylor polynomial for the given function \(f(x)\) about \(x=0 .\) Compare the bound with the actual error. $$\ln (1.5), f(x)=\ln (1+x)$$

Construct the first three Fourier approximations to the square wave function $$f(x)=\left\\{\begin{array}{rr} -1 & -\pi \leq x<0 \\\1 & 0 \leq x<\pi\end{array}\right.$$ Use a calculator or computer to draw the graph of each approximation.

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\arcsin x$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\cos x, \quad n=2,4,6$$

Find the first four nonzero terms of the Taylor series for the function about 0. $$\frac{1}{1-x}$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$t \sin (3 t)$$

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\ln (1+x), \quad n=5,7,9$$

Find the first four nonzero terms of the Taylor series for the function about 0. $$\frac{1}{\sqrt{1+x}}$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\frac{1}{\sqrt{1-z^{2}}}$$

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

Find the first four nonzero terms of the Taylor series for the function about 0. $$\sqrt[3]{1-y}$$

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. $$f(x)=x^{2}, \quad-\pi

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\frac{z}{e^{z^{2}}}$$

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

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

(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 \(n^{\text {th }}\). 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. $$\phi^{3} \cos \left(\phi^{2}\right)$$

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

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. $$g(x)=x, \quad-\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 \(a\). $$\cos t, \quad a=\pi / 6$$

(a) Let \(f(x)=e^{x} .\) Find a bound on the magnitude of the error when \(f(x)\) is approximated using \(P_{3}(x)\) its Taylor approximation of degree 3 around 0 over the interval [-2,2] (b) What is the actual maximum error in approximating \(f(x)\) by \(P_{3}(x)\) over the interval [-2,2]\(?\)

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 \(x\).

Find the Taylor series about 0 for the functions including the general term. $$(1+x)^{3}$$

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\). $$\sqrt{1-x}, \quad a=0, \quad n=3$$

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

Let \(f(x)=\cos x\) and let \(P_{n}(x)\) be the Taylor approximation of degree \(n\) for \(f(x)\) around \(0 .\) Explain why, for any \(x,\) we can choose an \(n\) such that $$\left|f(x)-P_{n}(x)\right|<10^{-9}$$

Find the Taylor series about 0 for the functions including the general term. $$t \sin \left(t^{2}\right)-t^{3}$$

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\). $$e^{x}, \quad a=1, \quad n=4$$

Find the Taylor series about 0 for the functions including the general term. $$\frac{1}{\sqrt{1-y^{2}}}$$

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\). $$\frac{1}{1+x}, \quad a=2, \quad n=4$$

Find the first four terms of the Taylor series for the function about the point \(a\). $$1 / x, \quad a=1$$