91Ó°ÊÓ

The Taylor polynomial of degree 7 of \(f(x)\) is given by $$ P_{7}(x)=1-\frac{x}{3}+\frac{5 x^{2}}{7}+8 x^{3}-\frac{x^{5}}{11}+8 x^{7} $$ Find the Taylor polynomial of degree 3 of \(f(x)\)

Find an expression for the general term of the series and give the range of values for the index \((n \text { or } k\) for example). $$\frac{1}{1+x}=1-x+x^{2}-x^{3}+x^{4}-\cdots$$

expand the quantity about 0 in terms of the variable given. Give four nonzero terms. $$\frac{1}{(a+r)^{2}} \text { in terms of } \frac{r}{a}$$

The function \(f(x)\) is approximated near \(x=0\) by the third-degree Taylor polynomial $$ P_{3}(x)=2-x-x^{2} / 3+2 x^{3} $$ Give the value of (a) \(\quad f(0)\) (b) \(f^{\prime}(0)\) (c) \(\quad f^{\prime \prime}(0)\) (d) \(\quad f^{\prime \prime \prime}(0)\)

expand the quantity about 0 in terms of the variable given. Give four nonzero terms. \(\sqrt[3]{P+t}\) in terms of \(\frac{t}{P}\)

expand the quantity about 0 in terms of the variable given. Give four nonzero terms. $$\frac{a}{\sqrt{a^{2}+x^{2}}} \text { in terms of } \frac{x}{a}, \text { where } a>0$$

Find the second-degree Taylor polynomial for \(f(x)=\) \(4 x^{2}-7 x+2\) about \(x=0 .\) What do you notice?

Find an expression for the general term of the series and give the range of values for the index \((n \text { or } k\) for example). $$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\cdots$$

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\sqrt{(1+t)} \sin t$$

For \(|x| \leq 0.1,\) graph the error $$E_{0}=\cos x-P_{0}(x)=\cos x-1$$ Explain the shape of the graph, using the Taylor expansion of \(\cos x .\) Find a bound for \(\left|E_{0}\right|\) for \(|x| \leq 0.1\).