The Natural Exponential Function

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

Why use such a strange base for an exponential function?

• 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) = ex

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 = 2x and y = 3x.

Natural Exponential Function

Scientific calculators have a special key for the function f(x) = ex.

• We use this key in the next example.

E.g. 6—Evaluating the Exponential Function

Evaluate each expression correct to five decimal places.

(a) e3

(b) 2e–0.53

(c) e4.8

E.g. 6—Evaluating the Exponential Function

We use the ex key on a calculator to evaluate the exponential function.

(a) e3 ≈ 20.08554

(b) 2e–0.53 ≈ 1.17721

(c) e4.8 ≈ 121.51042

E.g. 7—Transformations of the Exponential Function

Sketch the graph of each function.

(a) f(x) = e–x

(b) g(x) = 3e0.5x

E.g. 7—Transformations

We start with the graph of y = ex and reflect in the y-axis to obtain the graph of y = e–x.

Example (a)

E.g. 7—Transformations

We calculate several values, plot the resulting points, and then connect the points with a smooth curve.

Example (b)

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:

0.97

10,000( )5 1245 tv t

e

(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.

E.g. 8—An Exponential Model for the Spread of a Virus

E.g. 8—Spread of Virus

• We conclude that 8 people initially have the disease.

Example (a)

0(0) 10,000 /(5 1245 )

10,000 /1250

8

v e

E.g. 8—Spread of Virus

Using a calculator, we evaluate v(1), v(2), and v(5).

Then, we round off to obtain these values.

Example (b)

E.g. 8—Spread of Virus

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.

Example (c)

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

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

In general, after k periods, the amount is:

A = P(1 + 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

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

( ) 1n trA t P

n

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

/

( ) 1

1

11

n t

r tn r

r tm

rA t Pn

rPn

Pm

Compound Interest

Recall that, as m becomes large, the quantity (1 + 1/m)m approaches the number e.

• Thus, the amount approaches A = Pert.

• This expression gives the amount when the interest is compounded at “every instant.”

Continuously Compounded Interest

Continuously compounded interest is calculated by

A(t) = Pert

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) = 1000e(0.12)3 = 1000e0.36

= $1433.33

• Compare this amount with the amounts in Example 9.