Chapter 5, Problem 19PS

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Approximating a Function(a) Use a graphing utility to compare the graph of the function y = e x with the graph of each given function.(i) y 1 = 1 + x 11 (ii) y 2 = 1 + x 11 + x 2 21 (iii) y 3 = 1 + x 11 + x 2 21 + x 3 31 (b) Identify the pattern of successive polynomials in part (a), extend the pattern one more term, and compare the graph of the resulting polynomial function with the graph of y = e x (c) What do you think this pattern implies?

(a)

To determine

To graph: The following functions; y=ex and y1=1+x1!,y2=1+x1!+x22!and y3=1+x1!+x22!+x33! with the use of graphing utility and compare them.

Explanation

Given:

The functions: y=ex,â€‰â€‰y1=1+x1!,â€‰â€‰y2=1+x1!+x22!â€‰â€‰andÂ y3=1+x1!+x22!+x33!.

Graph:

First of all, sketch the functions y=ex and y1 in the same window for 0<x<3 by

TI83 calculators:

Step 1: Press the [Y=] key, then there will appear the equations for y.

Step 2: Enter the equations in Y1.Here, Y1=e^(X).

Step 3: Enter the equations in Y2.Here, Y2=1+X.

Step 4: Press [WINDOW] and then edit the values as

Xmin=0Xmax=3XScale=.5Ymin=0

and

Ymax=20YScale=1

Step 7: Press the [GRAPH] key to plot the graph.

Now, sketch the functions y=ex and y2 in the same window for 0<x<3 by

TI83 calculators:

Step 1: Press the [Y=] key, then there will appear the equations for y

(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=ex.

(c)

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

y1=1+x1!,y2=1+x1!+x22!and y3=1+x1!+x22!+x33!.

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