### Solving Problems Using Newton's Method

Solving algebraic equations is a common exercise in introductory Mathematics classes. However, sometimes equations cannot be solved using simple algebra and we might be required to find a good, accurate $estimate$ of the exact solution. A common and easily used algorithm to find a good estimate to an equation's exact solution is Newton's Method (also called the Newton-Raphson Method), which was developed in the late 1600's by the English Mathematicians Sir Isaac Newton and Joseph Raphson .

The algorithm for Newton's Method is simple and easy-to-use. It uses the the first derivative of a function and is based on the basic Calculus concept that the derivative of a function $f$ at $x=c$ is the slope of the line tangent to the graph of $y=f(x)$ at the point $(c, f(c))$. Let's carefully construct Newton's Method.

Let $y=f(x)$ be a differentiable function. Our goal is to solve the equation $f(x)=0$ for $x$. Let's call the exact solution to this equation $x=r$. See the diagram below.

We begin with an $initial \ guess$ $x_{0}$. At the point $(x_{0}, f(x_{0}))$ draw the tangent line. Denote $x_{1}$ as the point where this tangent line crosses the $x$-axis. The point $x_{1}$ is our second guess. Repeat. At the point $(x_{1}, f(x_{1}))$ draw the tangent line. Denote $x_{2}$ as the point where this tangent line crosses the $x$-axis. The point $x_{2}$ is our third guess. Repeat ... etc. See the diagram below.

We can see that these successive guesses, $\ x_{0}, x_{1}, x_{2}, x_{3}, \cdots \$ zig-zag their way and get closer and closer to the exact solution $x=r$. Let's create a $recursion$ which will generate a sequence of successive guesses. Let's first consider how we can go from one guess to the next, i.e., from guess $x_{n}$ to guess $x_{n+1}$. See the diagram below.

The $SLOPE$ of the tangent line at the point $(x_{n}, f(x_{n}))$ is $$I.) \ \ \ m = f'(x_{n})$$ and $$II.) \ \ \ m = \displaystyle{ rise \over run } = { f(x_{n}) \over x_{n} - x_{n+1} }$$ Now set the two slopes equal to each other getting $$f'(x_{n}) = { f(x_{n}) \over x_{n} - x_{n+1} } \ \ \ \ \longrightarrow$$ $$x_{n} - x_{n+1} = { f(x_{n}) \over f'(x_{n})} \ \ \ \ \longrightarrow$$ $$x_{n+1} = x_{n} - { f(x_{n}) \over f'(x_{n})} \ \ \ \ \longrightarrow$$ i.e., the recursion for Newton's Method is $$x_{n+1} = x_{n} - { f(x_{n}) \over f'(x_{n})}$$

In the list of Newton's Method Problems which follows, most problems are average and a few are somewhat challenging. I recommend using the Desmos Graphing Calculator if you want to graph functions. It's fun and easy to use.

• PROBLEM 1 : Apply Newton's Method to the equation $x^3+x-5=0$. Begin with the given initial guess, $x_{0}$, and find $x_{1}$ and $x_{2}$. Then use a spreadsheet or some other technology tool to find the solution to this equation to five decimal places.

a.) Use the initial guess $x_{0}=0$.
b.) Use the initial guess $x_{0}=1$.
c.) Use the initial guess $x_{0}=-1$.

• PROBLEM 2 : Apply Newton's Method to the equation $x^3=x^2+2$. Begin with the given initial guess, $x_{0}$, and find $x_{1}$ and $x_{2}$. Then use a spreadsheet or some other technology tool to find the solution to this equation to five decimal places.

a.) Use the initial guess $x_{0}=1$.
b.) Use the initial guess $x_{0}=2$.
c.) Use the initial guess $x_{0}=0.5$.
d.) Use the initial guess $x_{0}=1000$ !
e.) Use the initial guess $x_{0}=-500$ !

• PROBLEM 3 : Use Newton's Method to estimate the value of $\ 100^{1/5} \$ to eight decimal places.

• PROBLEM 4 : Use Newton's Method to estimate the value of $\pi$ to ten decimal places. NOTE: There are many possible functions to use which will lead to a correct solution.

• PROBLEM 5 : Apply Newton's Method to the equation $f(x)=0$, where function $f$ is given below. Start with an initial guess of $x_{0}=h>0$. $$f(x) = \cases{ \sqrt{x} , \ if \ x \ge 0 \cr -\sqrt{-x} , \ if \ x<0 \cr }$$