극한·미분 심화 전체 문제

For the following exercises, start at

a. ${x}_{0}=0.6$ and

b. ${x}_{0}=2.$

Compute ${x}_{1}$ and ${x}_{2}$ using the specified iterative method.

${x}_{n+1}={x}_{n}{}^{2}+{x}_{n}-2$

For the following exercises, start at

a. ${x}_{0}=0.6$ and

b. ${x}_{0}=2.$

Compute ${x}_{1}$ and ${x}_{2}$ using the specified iterative method.

${x}_{n+1}=\frac{1}{2}{x}_{n}-1$

For the following exercises, start at

a. ${x}_{0}=0.6$ and

b. ${x}_{0}=2.$

Compute ${x}_{1}$ and ${x}_{2}$ using the specified iterative method.

${x}_{n+1}=|{x}_{n}|$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

${x}^{2}-10=0$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

${x}^{4}-100=0$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

${x}^{2}-x=0$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

${x}^{3}-x=0$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

$x+5\cos (x)=0$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

$x+\tan (x)=0,$ choose ${x}_{0}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

$\frac{1}{1-x}=2$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

$1+x+{x}^{2}+{x}^{3}+{x}^{4}=2$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

${x}^{3}+{(x+1)}^{3}={10}^{3}$

For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess ${x}_{0}$ that is not the exact root.

$x={\sin }^{2}(x)$

For the following exercises, use Newton’s method to find the fixed points of the function where $f(x)=x;$ round to three decimals.

$\sin x$

For the following exercises, use Newton’s method to find the fixed points of the function where $f(x)=x;$ round to three decimals.

$\tan (x)$ on $x=\left(\frac{\pi }{2},\frac{3\pi }{2}\right)$

For the following exercises, use Newton’s method to find the fixed points of the function where $f(x)=x;$ round to three decimals.

${e}^{x}-2$

For the following exercises, use Newton’s method to find the fixed points of the function where $f(x)=x;$ round to three decimals.

$\ln (x)+2$

To find candidates for maxima and minima, we need to find the critical points ${f}^{'}(x)=0.$ Show that to solve for the critical points of a function $f(x),$ Newton’s method is given by ${x}_{n+1}={x}_{n}-\frac{{f}^{'}({x}_{n})}{{f}^{''}({x}_{n})}.$

What additional restrictions are necessary on the function $f?$

For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.

Minimum of $f(x)={x}^{2}+2x+4$