Newton’s Method: Fresh Take

  • 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.