Chapter 3, Problem 8PS

To determine

To identify: the pattern of successive polynomials. Extend the pattern one more term and compare the graph of resulting polynomial function with the graph of $y=e^x$ .

Answer to Problem 8PS

By increasing one term the graph is coming close to the graph $y=e^x$ .

Explanation of Solution

Given information:

Consider the following expressions

  $\begin{array}{l}y=e^x\\ y_1=1+\frac{x}{1!}+\frac{x^2}{2!}\\ y_2=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}\\ y_3=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\end{array}$

Calculation:

Use the T1-83 calculator.

Enter the expression one by one in the T1-83 calculator as shown below

  

Click on the graph button and below is the graph of the expression $y=e^x$

  

Graph of the expression $y_1=1+\frac{x}{1!}+\frac{x^2}{2!}$

  

Graph of the expression $y_2=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}$

  

Graph of the expression $y_3=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}$

  

All the graphs in one axis

  

Observing all the graphs, it is clear that by increasing one term the graph is coming close to the graph $y=e^x$ .