For $f(x)=x^2-10$ with $x_0=3$, find $x_1$ to three decimal places using Newton’s Method.

For $f(x)=x^3-9$ with $x_0=2$, find $x_1$ to three decimal places.

For $f(x)=\sin x-\frac{1}{2}$ with $x_0=0.5$ radians, find $x_1$ to three decimal places.

For $f(x)=\ln x-2$ with $x_0=7$, find $x_1$ to three decimal places.

For $f(x)=x^2+x-6$ with $x_0=1.5$, find $x_1$ to three decimal places.

For $f(x)=e^x-4$ with $x_0=1.5$, find $x_1$ to three decimal places.

For $f(x)=x^3-5$ with $x_0=1.5$, find $x_2$ to three decimal places.

For $f(x)=x^3+x-1$ with $x_0=1$, find $x_2$ to three decimal places.

For $f(x)=x^2+2x-4$ with $x_0=1$, find $x_2$ to three decimal places.

For $f(x)=x^3-2x-5$ with $x_0=2$, find $x_2$ to three decimal places.

For $f(x)=\ln x+x-3$ with $x_0=2$, find $x_2$ to three decimal places.

For $f(x)=e^{-x}-x$ with $x_0=0.5$, find $x_2$ to three decimal places.

Which formula is the Newton iteration for $f(x)=x^2-4x+1$?

Write the Newton iteration formula for $f(x)=e^x-3x$.

Which Newton iteration matches $f(x)=\ln x+x^2-5$?

Which formula gives Newton’s Method for $f(x)=\sin (2x)-x$?

For $f(x)=x^3-3x+1$, why does Newton’s Method fail at $x_0=1$?

For $f(x)=x^3-2x+2$ with $x_0=0$, what problem occurs in Newton’s Method?

For $f(x)=\sqrt[3]{x}$ with $x_0=1$, why does Newton’s Method diverge from the root $0$?

For $f(x)=x^2+1$ with $x_0=1$, what happens when Newton’s Method is continued to $x_2$?

A continuous differentiable $f$ has $f(-2)=5$, $f(-1)=-1$, $f(0)=-2$, and $f(1)=3$. Choose a start for the root bracketed between $0$ and $1$.

A graph of differentiable $f$ shows roots between $-1$ and $0$ and between $4$ and $5$; the tangent near $x=0$ is almost horizontal. Choose $x_0$ for the root between $4$ and $5$.

For $f(x)=x^2-1$ with $x_0=0.1$, which root do the Newton iterates approach?

A table gives $f(2)=-6$, $f(3)=-2$, $f(4)=5$, and $f(5)=9$ for a continuous differentiable $f$. Choose $x_0$ to seek the bracketed root.