Chapter 4.7, Problem 4E

For each initial approximation, determine graphically what happens if Newton’s method is used for the function whose graph is shown.

(a) x₁ = 0

(b) x₁ = 1

(c) x₁ = 3

(d) x₁ = 4

(e) x₁ = 5

To determine

To sketch: The graph of a tangent line at $x_1=0$.

Explanation of Solution

At first draw the tangent line at the point $x_1=0$ and the tangent line intercept the $x$ -axis.

Let the intercept point is $x_2$ and the value of that point is negative.

Similarly draw the tangent line at the point $x_2$ and the tangent line intercept the $x$-axis.

Let that intercept point is $x_3$ and value of that point more negative than previous one.

Graph:

Observation:

From Figure 1, observed that the point $x_1=0$ is Newton’s method fail.

To determine

To sketch: The graph of a tangent line at x₁ = 1.

Explanation of Solution

At first draw the graph of tangent line at the point $x_1=1$ and the tangent line intercept $x$-axis.

Let the intercept point is $x_2$ and that is parallel to $x$-axis.

Graph:

Observation:

From Figure 2 observed that the Newton’s method fail.

To determine

To sketch: The graph of a tangent line at x₁ = 3.

Explanation of Solution

At first draw the tangent line at the point $x_1=4$ and the tangent line intercept the $x$ -axis.

Let the intercept point is $x_2$.

Graph:

Observation:

From Figure 3, observed that the point $x_2=1$ The Newton’s method fail.

To determine

To sketch: The graph of a tangent line at x₁ = 4.

Explanation of Solution

At first draw the graph of tangent line at the point $x_1=1$ and the tangent line intercept $x$-axis.

Let the intercept point is $x_2$ and that is parallel to $x$-axis.

Graph:

Observation:

From Figure 4 observed that the Newton’s method fail.

To determine

To sketch: The graph of a tangent line at x₁ = 5.

Explanation of Solution

At first draw the tangent line at the point $x_1=5$ and the tangent line intercept the $x$ -axis.

Let the intercept point is $x_2$.

Similarly draw the tangent line at the point $x_2$ and the tangent line intercept the $x$-axis.

Let that intercept point $x_3$.

Graph:

Observation:

From Figure 5, observed that the point $x_1=5$ is Newton’s method work.