91Ó°ÊÓ

Use a graphing calculator program for Newton's method to approximate the root of each equation beginning with the given \(x_{0}\) and continuing until two successive approximations agree to nine decimal places. $$ \begin{array}{l} x^{5}+5 x-4=0 \\ x_{0}=1 \end{array} $$

Find the sum of each finite geometric series. $$ 1+2+2^{2}+2^{3}+\cdots+2^{10} $$

Find the Taylor series at \(x=0\) for each function by calculating three or four derivatives and using the definition of Taylor series. \(\frac{1}{3-x}\)

Find the Taylor series at \(x=0\) for each function by calculating three or four derivatives and using the definition of Taylor series. \(\sqrt{2 x+1}\)

Find the fifth Taylor polynomial for \(\sin 2 x\) by taking the fifth Taylor polynomial for \(\sin x\) (page 661 ) and replacing \(x\) by \(2 x\).

Use a graphing calculator program for Newton's method to approximate the root of each equation beginning with the given \(x_{0}\) and continuing until two successive approximations agree to nine decimal places. $$ \begin{array}{l} x+\ln x-5=0 \\ x_{0}=2 \end{array} $$

Find the sum of each finite geometric series. $$ 3+3 \cdot 4+3 \cdot 4^{2}+\cdots+3 \cdot 4^{5} $$

Use a graphing calculator program for Newton's method to approximate the root of each equation beginning with the given \(x_{0}\) and continuing until two successive approximations agree to nine decimal places. $$ \begin{array}{l} x^{2}+\ln x-3=0 \\ x_{0}=2 \end{array} $$

Find the Taylor series at \(x=0\) for each function by calculating three or four derivatives and using the definition of Taylor series. \(2 x^{2}-7 x+3\)

Find the sum of each finite geometric series. $$ 2+2 \cdot 3+2 \cdot 3^{2}+\cdots+2 \cdot 3^{9} $$