The function .

Apply derivative on each side with respect to .

(a) Graph of the function :

(b)

.

Perform Newton approximation for .

Observe the table,

Zero of the function is .

(c)

Perform Newton approximation for .

Observe the table,

Zero of the function is .

(d)

The points  and .

 and .

Find the tangent lines.

At the point :

.

At the point :

.

Graph the tangent lines.

Observe the graph :

The -intercept of  is .

The -intercept of  is .

The -intercepts correspond to the values resulting from the first iteration of Newton s method.

(e) If the initial estimate  is not sufficiently close to the desired zero of a function, the -intercept of the corresponding tangent line to the function may approximate a second zero of the function.

(a) Graph of the function  :

(b) Zero of the function is .

(c) Zero of the function is .

(d) The -intercept of  is .

The -intercept of  is .

The -intercepts correspond to the values resulting from the first iteration of Newton s method.

(e) If the initial estimate  is not sufficiently close to the desired zero of a function, the -intercept of the corresponding tangent line to the function may approximate a second zero of the function.