1. 1 6.11 Newton’s Method Problem: need to solve an equation of the form f(x)=0. Graphically, the solutions correspond to the points of intersection of the curve y=f(x) with the x-axis. Approach: to find the intercept, we approximate the curve by its tangents 1.Make a rough estimate to define an initial point. 2.Find the equation of the tangent line to the curve at this point. 3.Evaluate the intercept of the tangent line (i.e. approximate the curve by its tangent). 4.Use the intercept of the tangent line as a new (finer) estimate and repeat the procedure.

2. 2 Newton’s Method (cntd) Take initial point The equation for the tangent line: Intercept of the tangent line is a new estimate…

3. 3 Example: Use the Newton’s method to find a root of the following equation to six decimal places Solution: 1.Estimate the given function at a few points and take x=1 as the first estimate. 2.Find the derivative of the function: 3.Evaluate the derivative and the function at the chosen point and find the next approximation: 4.Use as the next approximation…

4. 4 Homework Section 6.11: 7,15.