- Explain how Newton’s method uses repetition to find roots of equations
- Recognize when Newton’s method does not work
- Apply methods that repeat steps to solve different types of mathematical problems
Approximating with Newton’s Method
The Main Idea
- Purpose:
- Efficiently approximate roots of equations in the form [latex]f(x) = 0[/latex]
- Useful when analytical solutions are difficult or impossible to find
- The Method:
- Start with an initial guess [latex]x_0[/latex]
- Iteratively improve the approximation using the formula: [latex]x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}[/latex]
- Geometric Interpretation:
- Each iteration finds the [latex]x[/latex]-intercept of the tangent line at the current approximation
- Convergence:
- Typically converges quickly to a root when successful
- May require multiple iterations for desired accuracy
- Potential Failures:
- Derivative equals zero at an approximation point
- Convergence to an unintended root
- Failure to converge (e.g., oscillating between values)
Use Newton’s method to approximate [latex]\sqrt{3}[/latex] by letting [latex]f(x)=x^2-3[/latex] and [latex]x_0=3[/latex]. Find [latex]x_1[/latex] and [latex]x_2[/latex].