(a)

The function is .

Draw a coordinate plane.

Graph the function .

Graph :

(b)

The function is .

Differentiate on each side with respect to .

.

Newtons approximation method formula : .

Perform newton approximation for .

.

Perform newton approximation for .

. 

Perform newton approximation for .

.

Perform newton approximation for .

.

Thus, the zero of  is .

(c)

The function is .

Differentiate on each side with respect to .

.

Newtons approximation method formula : .

Perform newton approximation for .

.

Perform newton approximation for .

.

Perform newton approximation for .

.

Thus, the zero of  is .

Observe that, the two results are different.

(d)

The points are  and .

The function is .

 and .

Now, the points are  and .

Find the tangent line to  at .

The function is .

Differentiate on each side with respect to .

.

This is the slope of the tangent line.

Slope intercept form of lime equation is , where  is slope and  is  - intercept.

.

Find the  - intercept by substituting the point  in .

Thus, the tangent line is .

Find the tangent line to  at .

The function is .

Differentiate on each side with respect to .

.

This is the slope of the tangent line.

Slope intercept form of lime equation is , where  is slope and  is  - intercept.

.

Find the  - intercept by substituting the point  in .

Thus, the tangent line is .

Draw a coordinate plane.

Graph the function  and graph the tangent lines  and  in the same window.

Graph :

Observe the above graph :

The - intercept of the tangent line  is .

The - intercept of the tangent line  is .

Therefore, the - interceps and the first iteration of Newtons method using the respective initial guesses are equal.

(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)

The zero of  is .

(c)

The zero of  is .

(d)

Graph :

The - intercept of the tangent line  is .

The - intercept of the tangent line  is .

Therefore, the - interceps and the first iteration of Newtons method using the respective initial guesses are equal.

(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.