## Newton's Method for Finding Roots

### Practice Problem Answers

**Practice 1:** so and the slope of the tangent line at the point is . Using the point-slope form for the equation of a line, the equation of the tangent line is or .

The -coordinate of a point on the -axis is 0 so we need to put and solve the linear equation for so .

The line tangent to the graph of at the point intersects the -axis at the point .

**Practice 2: **The approximate locations of and are shown in Fig. 20.

**Practice 4: **Fig. 21 shows the first iteration of Newton's Method for , and .

If , the iterates approach the root at .

If , the iterates approach the root at .

If , the iterates approach the root at .

The graph of the cube root has a shape similar to Fig. 14, and the behavior of the iterates is similar to the pattern in that figure. Unless (the only root of ) the iterates alternate in sign and double in magnitude with each iteration: they get progressively farther from the root with each iteration.