Chapter Contents

• Applications of Differentiation
• 1. Tangents and Normals
• 2. Newton's Method
• 3. Curvilinear Motion
• 4. Related Rates
• 5. Curve Sketching Using Differentiation
• 6. More Curve Sketching Using Differentiation
• 7. Applied Maximum and Minimum Problems
• 8. Radius of Curvature
• Applications of Differentiation Problem Solver

• Comments, Questions?

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2. Newton's Method for Solving Equations

by M. Bourne

Computers use iterative methods to solve equations. The process involves making a guess at the true solution and then applying a formula to get a better guess and so on until we arrive at an acceptable approximation for the solution.

If we wish to find x so that `f(x) = 0` (a common type of problem), then we guess some value x₁ which is close to the desired solution and then we get a better approximation using Newton's Method:

`x_2=x_1-(f(x_1))/(f^'(x_1)`

[This is just based on the point-slope form of a straight line].

Example 1

Find the root of

2x² − x − 2 = 0

between `1` and `2`.

Answer

Functions with Multiple Roots

Many functions have multiple roots, so you need to understand what is going on and give the computer a guess close to your desired answer.

Example 2

Solve 1− t² + 2^t = 0

[Scientific Notebook cannot find the solution directly for us. We need to know how to properly use the tool to get the solution, either with graphs or setting up Newton's Method. This could involve giving an initial estimate for the root.]

Answer