Chapter 3.2, Problem 125E

(a).

To determine

To compare: the graph of the function $y=\ln x$ with the graph of each given function.

Explanation of Solution

Given:

The given functions:

  $\begin{array}{l}{y}_1=x-1\\ $ y_2=(x-1)-\frac{1}{2}{(x-1)}^2$\end{array}$

  $y_3=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3$

Graph:

As per the given problem

The given functions:

  $\begin{array}{l}y=\ln x\\ y_1=x-1\\ $ y_2=(x-1)-\frac{1}{2}{(x-1)}^2$\end{array}$

  $y_3=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3$

Sketch the graphs using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions, which is required to graph:

  $\begin{array}{l}y=\ln x\\ y_1=x-1\\ $ y_2=(x-1)-\frac{1}{2}{(x-1)}^2$\end{array}$

  $y_3=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3$

Step 4: Press GRAPH button to graph the functions.

  

Interpretation: All the functions intersect at $(1,0)$ .

(b).

To determine

To identify: the pattern of successive polynomials in part (a).

Extend the pattern one more term and compare the graph of the resulting polynomial with the graph of the function $y=\ln x$ a and draw conclusions.

Answer to Problem 125E

The successive polynomials is of the form:

  $y_n={\displaystyle \sum _n\frac{{(-1)}^{n-1}}{n}{(x-1)}^n}$ , where $n=1,\text{ 2, 3, 4}\mathrm{...}$

  $y_4=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3-\frac{1}{4}{(x-1)}^4$

  

The graph of function $y_4$ is similar to an inverted parabola. It can be concluded that for even values of n, the graph of the equation is similar to an inverted parabola.

Explanation of Solution

Given:

The given functions:

  $\begin{array}{l}{y}_1=x-1\\ $ y_2=(x-1)-\frac{1}{2}{(x-1)}^2$\end{array}$

  $y_3=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3$

As per the given problem,

Consider the given functions.

The polynomials are increasing in the pattern

  $y_n={\displaystyle \sum _n\frac{{(-1)}^{n-1}}{n}{(x-1)}^n}$

Where $n=1,\text{ 2, 3, 4}\mathrm{...}$

Extending the polynomial for $n=\text{4}$

  $y_4=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3-\frac{1}{4}{(x-1)}^4$

Sketch the graphs using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions, $y=\ln x$ and $y_4=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3-\frac{1}{4}{(x-1)}^4$ , which is required to graph.

Step 4: Press GRAPH button to graph the functions.

  

The graph of function $y_4=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3-\frac{1}{4}{(x-1)}^4$ is similar to an inverted parabola. It can be concluded that for even values of n, the graph of the equation is similar to an inverted parabola.