- 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 f(x)=0f(x)=0
- Useful when analytical solutions are difficult or impossible to find
- The Method:
- Start with an initial guess x0x0
- Iteratively improve the approximation using the formula: xn=xn−1−f(xn−1)f′(xn−1)
- Geometric Interpretation:
- Each iteration finds the x-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 √3 by letting f(x)=x2−3 and x0=3. Find x1 and x2.