The Natural Exponential Function (4.3)

An elaboration on compounded growth

A POD to warm up

Find the amount of money you’d have at the end of one year in an account if you invested $1000 at 6% interest, compounded

annually

monthly

weekly

hourly

by the minute

A POD to warm up

Find the amount of money you’d have in an account if you invested $1000 at 6% interest, compounded

annuallymonthlyweeklyhourlyby the minute

Notice the limit to growth as we approach continuous compounding.

Consider another limit

What is the value of

if n=1

if n=10

if n=100

if n=1000

if n=100,000

if n=1,000,000

11

n

n

Consider another limit

As n approaches infinity, the value of the expression approaches a limit. That limit is defined as e,, the base for the natural exponential and logarithmic functions.

e is an irrational, transcendental number, much like pi.

It’s numerical value can be rounded to 2.71828…

11

n

n

Consider another limit

In other words:

e

and the natural exponential function isf(x)=ex.

Graph y=ex on your calculators. How is it similar to the graph of y=3x?

n lim 1

1

n

n

2.71828...

Continuous compounding

There is such a thing as continuous compounding; its formula looks somewhat similar to the formula we used for compounded interest.

rtPeA

Continuous compounding

Let’s derive the formula. Start with our discrete compounding formula

Then, a little deliberate manipulation through replacement.

Let 1/k = r/n. Then n=kr. And if we substitute, we get

if k is infinitely large. This is the formula we use if we have continuous compounding.

A P 1r

n

nt

A P 11

k

krt

P 11

k

k

rt

Pert

Continuous compounding

Find the amount of money you’d have in the POD with continuous compounding.

How much would you have if you’d started with $10,000?

Natural exponential function

Just as with other bases of exponential functions, we can solve for the exponent as long as the bases are the same.

Solve for x: e3x=e2x-1.

Natural exponential function

We can also use it in applications (word problems).

In 1978, the population of blue whales in the southern hemisphere was thought to number 5000. With whaling outlawed and assuming an abundant food supply, the population N(t) is expected to grow exponentially according to the formula

N(t)=5000e.0036t.

where t is in years, and t=0 corresponds to 1978.

Why would t=0 correspond to 1978?Predict the population in the year 2010.