4.9 Newton’s method  (Page 5/7)

$f\left(x\right)=x^2+1$

$f\left(x\right)=x^3+2x+1$

$f(x)=\text{sin}x$

$f(x)=e^x$

$f(x)=x^3+3x{e}^x$

$f\left(x\right)=x^2-4,$ with $x_0=0$

$f\left(x\right)=x^2-4x+3,$ with $x_0=2$

What is the value of $\text{“}c\text{”}$ for Newton’s method?

$x_{n+1}=x_n{}^2-\frac{1}{2}$

$x_{n+1}=2x_n\left(1-x_n\right)$

$x_{n+1}=\sqrt{x_n}$

$x_{n+1}=\frac{1}{\sqrt{x_n}}$

$x_{n+1}=3x_n\left(1-x_n\right)$

$x_{n+1}=x_n{}^2+x_n-2$

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

$x_{n+1}=\left|x_n\right|$

$x^2-10=0$

$x^4-100=0$

$x^2-x=0$

$x^3-x=0$

$x+5\text{cos}\left(x\right)=0$

$x+\text{tan}\left(x\right)=0,$ choose $x_0\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

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

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

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

$x={\text{sin}}^2\left(x\right)$

$\text{sin}x$

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

$e^x-2$

$\text{ln}\left(x\right)+2$

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

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

Minimum of $f\left(x\right)=x^2+2x+4$

Minimum of $f\left(x\right)=3x^3+2x^2-16$

Minimum of $f\left(x\right)=x^2{e}^x$

Maximum of $f\left(x\right)=x+\frac{1}{x}$

Maximum of $f\left(x\right)=x^3+10x^2+15x-2$

Maximum of $f\left(x\right)=\frac{\sqrt{x}-\sqrt[3]{x}}{x}$

Minimum of $f\left(x\right)=x^2\text{sin}x,$ closest non-zero minimum to $x=0$

Minimum of $f\left(x\right)=x^4+x^3+3x^2+12x+6$

Newton’s method, $x^2+2=0$

Newton’s method, $0=e^x$

Newton’s method, $0=1+x^2$ starting at $x_0=0$

Solving $x_{n+1}=\text{−}{x}_n{}^3$ starting at $x_0=\mathrm{-1}$

Find a root to $0=x^2-x-3$ accurate to three decimal places.

Find a root to $0=\text{sin}x+3x$ accurate to four decimal places.

Find a root to $0=e^x-2$ accurate to four decimal places.

Find a root to $\text{ln}\left(x+2\right)=\frac{1}{2}$ accurate to four decimal places.

Why would you use the secant method over Newton’s method? What are the necessary restrictions on $f?$

$f\left(x\right)=x^2+2x+1,x_0=1$

$f\left(x\right)=x^2,x_0=1$

$f\left(x\right)=\text{sin}x,x_0=1$

$f\left(x\right)=e^x-1,x_0=2$

$f\left(x\right)=x^3+2x+4,x_0=0$

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 $\epsilon =0.25;$ 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 $\epsilon =0.8;$ 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.

The cost for printing a book can be given by the equation $C\left(x\right)=1000+12x+\left(\frac{1}{2}\right)x^{2\text{/}3}.$ Use Newton’s method to find the break-even point if the printer sells each book for $\text{\$}20.$