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.

Click HERE to return to the original list of various types of calculus problems.

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Duane Kouba ... September 20, 2020