Natural Exponential Function
Any positive number can be used as the base for an exponential function.
However, some are used more frequently than others.
• We will see in the remaining sections of the chapter that the bases 2 and 10 are convenient for certain applications.
• However, the most important is the number denoted by the letter e .

Number e
The number e is defined as the value that (1 + 1/ n ) n approaches as n becomes large.
• In calculus, this idea is made more precise through the concept of a limit.

Number e
The table shows the values of the expression (1 + 1/ n ) n for increasingly large values of n .
• It appears that, correct to five decimal places, e ≈ 2.71828

Number e
The approximate value to 20 decimal places is: e ≈ 2.71828182845904523536
• It can be shown that e is an irrational number.
• So, we cannot write its exact value in decimal form.

Number e
• It may seem at first that a base such as 10 is easier to work with.
• However, we will see that, in certain applications, it is the best possible base.

Natural Exponential Function —Definition
The natural exponential function is the exponential function f ( x ) = e x with base e .
• It is often referred to as the exponential function.

Natural Exponential Function
Since 2 < e < 3, the graph of the natural exponential function lies between the graphs of y = 2 x and y = 3 x .

Natural Exponential Function
f
x
e x
• We use this key in the next example.

E.g. 6 —Evaluating the Exponential Function
Evaluate each expression correct to five decimal places.
(a) e 3
(b) 2 e
–0.53
(c) e 4.8

E.g. 6 —Evaluating the Exponential Function
We use the e x key on a calculator to evaluate the exponential function.
(a) e 3 ≈ 20.08554
(b) 2 e
–0.53
≈ 1.17721
(c) e 4.8
≈ 121.51042

E.g. 7 —Transformations of the Exponential Function
Sketch the graph of each function.
(a) f ( x ) = e
– x
(b) g ( x ) = 3 e 0.5
x

E.g. 7 —Transformations Example (a)
We start with the graph of y = e x and reflect in the y -axis to obtain the graph of y = e
–x
.

E.g. 7 —Transformations Example (b)
We calculate several values, plot the resulting points, and then connect the points with a smooth curve.

E.g. 8 —An Exponential Model for the Spread of a Virus
An infectious disease begins to spread in a small city of population 10,000.
• After t days, the number of persons who have succumbed to the virus is modeled by:

10,000
 e

0.97
t

E.g. 8 —An Exponential Model for the Spread of a Virus
(a) How many infected people are there initially (at time t = 0)?
(b) Find the number of infected people after one day, two days, and five days.
(c) Graph the function v and describe its behavior.

v
E.g. 8 —Spread of Virus
  e 0 )
Example (a)

10,000 /1250

8
• We conclude that 8 people initially have the disease.

E.g. 8 —Spread of Virus Example (b)
Using a calculator, we evaluate v (1), v (2), and v (5).
Then, we round off to obtain these values.

E.g. 8 —Spread of Virus Example (c)
From the graph, we see that the number of infected people:
• First, rises slowly.
• Then, rises quickly between day 3 and day 8.
• Then, levels off when about 2000 people are infected.

Logistic Curve
This graph is called a logistic curve or a logistic growth model.
• Curves like it occur frequently in the study of population growth.

Compound Interest
Exponential functions occur in calculating compound interest.
• Suppose an amount of money P , called the principal, is invested at an interest rate i per time period.
• Then, after one time period, the interest is Pi , and the amount A of money is:
A = P + Pi + P (1 + i )

Compound Interest
If the interest is reinvested, the new principal is P (1 + i ), and the amount after another time period is:
A = P (1 + i )(1 + i ) = P (1 + i ) 2
• Similarly, after a third time period, the amount is:
A = P (1 + i ) 3

Compound Interest
k
A
P
i
k
• Notice that this is an exponential function with base 1 + i .

Compound Interest
Now, suppose the annual interest rate is r and interest is compounded n times per year.
Then, in each time period, the interest rate is i = r / n , and there are nt time periods in t years.
• This leads to the following formula for the amount after t years.

Compound Interest
Compound interest is calculated by the formula
( )

P



1
 r n


 n t where:
• A ( t ) = amount after t years
• P = principal
• t = number of years
• n = number of times interest is compounded per year
• r = interest rate per year

E.g. 9 —Calculating Compound Interest
A sum of $1000 is invested at an interest rate of 12% per year.
Find the amounts in the account after 3 years if interest is compounded:
• Annually
• Semiannually
• Quarterly
• Monthly
• Daily

E.g. 9 —Calculating Compound Interest
We use the compound interest formula with: P = $1000, r = 0.12, t = 3

Compound Interest
We see from Example 9 that the interest paid increases as the number of compounding periods n increases.
• Let’s see what happens as n increases indefinitely.

Compound Interest
If we let m = n / r , then
( )

P



1
 r n


 n t

P
 





1
 r n




P
 





1

1 m


 m
 

 r t
 

 r t

Compound Interest
Recall that, as m becomes large, the quantity (1 + 1/ m ) m approaches the number e .
• Thus, the amount approaches A = Pe rt .
• This expression gives the amount when the interest is compounded at “every instant.”

Continuously Compounded Interest
Continuously compounded interest is calculated by
A ( t ) = Pe rt where:
• A ( t ) = amount after t years
• P = principal
• r = interest rate per year
• t = number of years

E.g. 10 —Continuously Compounded Interest
Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.

E.g. 10 —Continuously Compounded Interest
We use the formula for continuously compounded interest with:
P = $1000, r = 0.12, t = 3
• Thus,
A (3) = 1000 e (0.12)3 = 1000 e 0.36
= $1433.33
• Compare this amount with the amounts in Example 9.