Chapters
Definition
The exponential function is one that assigns to each real value 
the power 
with 
and 
. This function is expressed as
The number 
is called the base.
Graphs of Exponential Functions
Let's study the behavior of the exponential function according to its base
We construct a value table for 
:
|  |
|---|---|
| -3 | 1/8 |
| -2 | 1/4 |
| -1 | 1/2 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
We draw the graph
Now we construct a value table for 
:
|  |
|---|---|
| -3 | 8 |
| -2 | 4 |
| -1 | 2 |
| 0 | 1 |
| 1 | 1/2 |
| 2 | 1/4 |
| 3 | 1/8 |
We draw the graph
We observe that the first function is strictly increasing, while the second is strictly decreasing; furthermore both are symmetric with respect to the 
axis.
Natural Exponential Function
This is denoted by 
where 
is given by
This notation was introduced by Leonhard Euler around 1730, when discovering many properties of this number. The number is irrational and its first ten decimal digits are 
.
Properties of the Exponential Function
1 Domain: 
.
2 Range: 
.
3 It is continuous.
4 The points 
and 
belong to the graph.
5 It is injective 
(no image has more than one preimage).
6 Increasing if 
.
7 Decreasing if 
.
8 The curves and 
are symmetric with respect to the axis.
9 The exponential function , with eventually grows faster than the power function 
for any 
.
10 The inverse function of the exponential function is 
. The inverse function of the natural exponential is 
.
Applications of the Exponential Function
Exponential functions are used to model a wide variety of phenomena such as population growth and interest rates.
Exponential Growth and Decay
The formula used to model population growth is given by
The function 
grows exponentially and represents the population quantity at time 
; 
represents the growth or decay constant; if 
it is called the growth constant, while if 
it is called the decay constant. 
represents the initial population at time zero, that is, 
.
The above formula is expressed in terms of the natural exponential, but sometimes it is expressed with base , this is simple to obtain, just apply the properties of exponents to 
and consider 
to obtain:
Example: A group of researchers study a bacterial culture. If at the beginning of the observation they have 
bacteria and half an hour later they have 
, find:
1 The number of bacteria after two hours.
2 The number of bacteria after three hours.
3 The average rate of change of the population during the second hour.
4 The time required to double the initial population.
5 When will the population equal 
?
To be able to answer what is requested, we first need to know the population growth formula with 
expressed in minutes.
We note that we know the initial population 
, but we lack the value of the growth constant. To find the value of 
we use the problem data: 
in the growth formula:
Dividing both sides by 
and applying the inverse function of the natural exponential, we obtain:
Thus the function that models the growth of the bacterial population is:
1 The number of bacteria after two hours is:
2 The number of bacteria after three hours:
3 The average rate of change of the population during the second hour
During the second hour, from time 
to 
, the population changed by 
, so the average rate in this time period is:
The population increases at an approximate average rate of 
bacteria per minute during the second hour.
4 The time required to double the initial population
For this we use the following equality
Dividing both sides by and applying the inverse function of the natural exponential, we obtain
Thus the time required for the bacterial population to double is 
minutes.
5 When will the population equal ?
For this we use the following equality
Dividing both sides by and applying the inverse function of the natural exponential, we obtain
Thus the time required for the bacterial population to be 
is 
minutes.
Compound Interest
An initial amount of money is invested at an interest rate 
expressed in decimals. If the interest is compounded only once, then the balance to obtain 
after adding the interest is:
If the interest is compounded more than once, the interest that is added to the account during one period will earn interest during the following periods. If the annual interest rate is and the interest is compounded times per year, then at the end of years, the interest was compounded 
times and the balance called future value is:
Example: If $500 is invested at a rate of 5% annually. Find the future value after 3 years if the interest is compounded quarterly.
To find the future value after 3 years if the interest is compounded quarterly, we use 
.
We substitute the values in the future value formula:
The balance obtained 
years is 
Continuously Compounded Interest
To know the balance of an investment at the end of years when the compounding frequency increases without limit, that is, the interest is not compounded quarterly, nor monthly, nor daily, but continuously, the formula is used
Example: If 
is invested at a rate of 
annually. Find the future value after 3 years if the interest is compounded continuously.
To find the future value after years if the interest is compounded continuously, we use 
.
We substitute the values in the future value formula:
The balance obtained after years is 
and is the upper limit for the possible balance.
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