Chapter 5, Problem 19PS

(a)

To determine

To graph: The following functions; $y=e^x$ and $y_1=1+\frac{x}{1!},y_2=1+\frac{x}{1!}+\frac{x^2}{2!}\text{and }{y}_3=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}$ with the use of graphing utility and compare them.

(b)

To determine

The pattern of successive polynomials in part (a) by extending the pattern by one more term, and also compare the graph of the resulting polynomial function with the graph of $y=e^x$.

(c)

To determine

The implication of the pattern on successive polynomials of the functions:

$y_1=1+\frac{x}{1!},y_2=1+\frac{x}{1!}+\frac{x^2}{2!}\text{and }{y}_3=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}$.