Exponential functions

The equation

defines the exponential function with base b . The domain is the set of all real numbers, while the range is the set of all positive real numbers ( y > 0). Note y cannot equal to zero.

( ) xf x b

Riddle Here is a problem related to exponential functions: Suppose you received a penny on the first day of

December, two pennies on the second day of December, four pennies on the third day, eight pennies on the fourth day and so on. How many pennies would you receive on December 31 if this pattern continues?

2) Would you rather take this amount of money or receive a lump sum payment of $10,000,000?

Solution (Complete the table)

Day No. pennies

1 1

2 2 2^1

3 4 2^2

4 8 2^3

5 16

6 32

7 64

Generalization Now, if this pattern continued, how many

pennies would you have on Dec. 31? Your answer should be 2^30 ( two raised to the

thirtieth power). The exponent on two is one less than the day of the month. See the preceding slide.

What is 2^30? 1,073,741,824 pennies!!! Move the decimal

point two places to the left to find the amount in dollars. You should get: $10,737,418.24

Solution, continued The obvious answer to question two is to take

the number of pennies on December 31 and not a lump sum payment of $10,000,000

(although, I would not mind having either amount!)

This example shows how an exponential function grows extremely rapidly. In this case, the exponential function

is used to model this problem.

( ) 2xf x

Graph of

Use a table to graph the exponential function above. Note: x is a real number and can be replaced with numbers such as as well as other irrational numbers. We will use integer values for x in the table:

( ) 2xf x

2

Table of values  

1 12

2

44

1 12

2 16

 

3 12

8

 

2 12

4

  

02 1

 

12 2

 

22 4

x y

-4

 -3

 -2

 -1

 0

 1

 2

Graph of y = ( ) 2xf x

Characteristics of the graphs of

where b> 1

1. all graphs will approach the x-axis as x gets large. 2. all graphs will pass through (0,1) (y-intercept) 3. There are no x – intercepts. 4. Domain is all real numbers 5. Range is all positive real numbers. 6. The graph is always increasing on its domain. 7. All graphs are continuous curves.

( ) xf x b

Graphs of if 0 < b < 1

1. all graphs will approach the x-axis as x gets large. 2. all graphs will pass through (0,1) (y-intercept) 3. There are no x – intercepts. 4. Domain is all real numbers 5. Range is all positive real numbers. 6. The graph is always decreasing on its domain. 7. All graphs are continuous curves.

( ) xf x b

Graph of Using a table of values once again, you will obtain the following graph.

The graphs of and will be symmetrical with respect to the y-axis, in general.

1( ) 2

2x

xf x

0

2

4

6

8

10

12

-4 -2 0 2 4

graph of y = 2 (̂-x)

approaches the positive x-axis as x gets large

passes through (0,1)

( ) xf x b ( ) xf x b

Graphing other exponential functions

Now, let’s graph

Proceeding as before, we construct a table of values and plot a few points.Be careful not to assume that the graph crosses the negative x-axis. Remember, it gets close to the x-axis, but never intersects it.

( ) 3xf x

Preliminary graph of ( ) 3xf x

Complete graph

0

5

10

15

20

25

30

-4 -2 0 2 4

Series1

y = 3 x̂

Other exponential graphs This is the graph of

It is symmetric to the graph of

with respect to the y-axis Notice that it is always

decreasing. It also passes through

(0,1).

( ) 4 xf x

( ) 4xf x

Exponential function with base e The table to the left

illustrates what happens to the expression

as x gets increasingly larger. As we can see from the table, the values approach a number whose approximation is 2.718

(1+1/x)^x

1 2

10 2.59374246

100 2.704813829

1000 2.716923932

10000 2.718145927

1000000 2.718280469

11

x

x

Leonard Euler Leonard Euler first demonstrated that

will approach a fixed constant we now call “e”. So much of our mathematical notation is due to Euler

that it will come as no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from "exponential", but it may have just be the next vowel after "a" and Euler was already using the notation "a" in his work. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731.

(http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html#s19)

11

x

x

Leonard Euler He made various discoveries

regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysis in infinitorum that he gave a full treatment of the ideas surrounding e. He showed that

e = 1 + 1/1! + 1/2! + 1/3! + ...

and that e is the limit of (1 + 1/n)^n as n tends to

infinity. Euler gave an approximation for e to 18 decimal places,

e = 2.718281828459045235

Graph of Graph is similar to

the graphs of

and

Has same characteristics as these graphs

graph of y = e^x

0

5

10

15

20

25

-4 -2 0 2 4

Series1

( ) xf x e

( ) 2xf x

( ) 3xf x

Growth and Decay applications The atmospheric pressure p

decreases with increasing height. The pressure is related to the number of kilometers h above the sea level by the formula:

Find the pressure at sea level ( h =1)

Find the pressure at a height of 7 kilometers.

0.145( ) 760 hP h e

Solution: Find the pressure at

sea level ( h =1) Find the pressure at

a height of 7 kilometers

0.145(1)(1) 760 657.42P e 0.145(7)(7) 760 275.43P e

Depreciation of a machine A machine is initially

worth dollars but loses 10% of its

value each year. Its value after t years is given by the formula

Find the value after 8 years of a machine whose initial value is $30,000

Solution:

0( ) (0.9 )tV t V

0V0

( ) (0.9 )tV t V

8(8) 30000(0.9 ) $12,914V

Compound interest The compound interest formula is

Here, A is the future value of the investment, P is the initial amount (principal), r is the annual interest rate as a decimal, n represents the number of compounding periods per year and t is the number of years

1nt

rA P

n

Problem: Find the amount to which $1500 will grow if deposited in

a bank at 5.75% interest compounded quarterly for 5 years.

Solution: Use the compound interest formula:

Substitute 1500 for P, r = 0.0575, n = 4 and t = 5 to obtain

=$1995.55

1nt

rA P

n

(4)(5)0.0575

1500 14

A