극한·미분 심화 전체 문제

For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method.

$f(x)={x}^{3}+2x+4,{x}_{0}=0$

Use Newton’s method to solve for the eccentric anomaly $E$ when the mean anomaly $M=\frac{\pi }{3}$ and the eccentricity of the orbit $\varepsilon =0.25;$ round to three decimals.

Use Newton’s method to solve for the eccentric anomaly $E$ when the mean anomaly $M=\frac{3\pi }{2}$ and the eccentricity of the orbit $\varepsilon =0.8;$ round to three decimals.

Use Newton’s method to determine the interest rate if the interest was compounded annually.

Use Newton’s method to determine the interest rate if the interest was compounded continuously.

For the following exercises, find the derivative $\frac{dy}{dx}.$

$y=\ln (2x)$

For the following exercises, find the derivative $\frac{dy}{dx}.$

$y=\ln (2x+1)$

For the following exercises, find the derivative $\frac{dy}{dx}.$

$y=\frac{1}{\ln \,x}$

For the following exercises, find the indefinite integral.

$\int \frac{dt}{3t}$

For the following exercises, find the indefinite integral.

$\int \frac{dx}{1+x}$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\frac{\ln (x)}{x}$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=x\,\ln (x)$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y={\log }_{10}x$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln (\sin \,x)$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln (\ln \,x)$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=7\,\ln (4x)$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln ({(4x)}^{7})$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln (\tan \,x)$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln (\tan (3x))$

For the following exercises, find the derivative $dy/dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y=\ln ({\cos }^{2}x)$