91Ó°ÊÓ

Use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-x)^{1.01} $$

Use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-2 x)^{2 / 3} $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \sqrt{x+2} \text { at } a=0 $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \sqrt{x+2} \text { at } a=1 $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \begin{array}{l} \text { 181. } \quad \sqrt{2 x-x^{2}} \quad \text { at } \quad a=1 \quad \text { (Hint: } \\ \left.2 x-x^{2}=1-(x-1)^{2}\right) \end{array} $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \sqrt{x} \text { at } a=4 $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \sqrt{x} \text { at } x=9 $$

Use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most \(1 / 1000\). \((15)^{1 / 4}\) using \((16-x)^{1 / 4}\)

Use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most \(1 / 1000\). \((1001)^{1 / 3}\) using \((1000+x)^{1 / 3}\)

Use the binomial approximation \(\sqrt{1-x} \approx 1-\frac{x}{2}-\frac{x^{2}}{8}-\frac{x^{3}}{16}-\frac{5 x^{4}}{128}-\frac{7 x^{5}}{256}\) for \(|x|<1\) to approximate each number. Compare this value to the value given by a scientific calculator. \(\frac{1}{\sqrt{2}}\) using \(x=\frac{1}{2}\) in \((1-x)^{1 / 2}\)