Exponentials

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Exponentials

Example of an Exponential Function

A biologist grows bacteria in a culture. If initially there were three grams of bacteria, after one day there are six grams of bacteria, and after two days, there are twelve grams, how many grams will there be at the end of the week?

Solution:

We draw a t chart

t	P(t)
0	3 = 3(2⁰)
1	6 = 3(2¹)
2	12 = 3(2²)

We see that the general formula is

P(t) = 3(2^t)

Hence after one week, we calculate

P(7) = 3(2⁷) = 384 grams of bacteria.

We call P(t) and exponential function with base 2.

Graphing Exponentials

Below is the graph of y = 2^x. It turns out that for any b > 1 the graph of y = b^t looks similar.

$Graph of y=2^x. It has a left horizontal asymptote, it goes through (0,1) and steeply goes up after.$

Notice that
1. the left horizontal asymptote at 0
2. The y-intercept is 1
3. The graph is always increasing.
Shifting techniques can also be used to graph variations of this curve.

Example

Graph y = 2^-x

Solution:

We see that the graph is reflected about the y-axis:

$Graph of y=2^(-x). It has a right horizontal asymptote, it goes through (0,1) and steeply goes up looking left from the y-axis.$

Three Properties of Exponents

b^xb^y = b^x+y
b^x/ b^y = b^x-y
(b^x)^y = b^xy

Definition

b^-x = 1 / b^x

Example

Simplify

3⁴(-3)^-1/[(3²)³]

Solution

3⁴(-3)^-1/[(3²)³] = 3⁴(-3)^-1/ 3⁶= -3⁴ /(3¹3⁶) = -3⁴/ 3⁷ = -1/3³ = -1/27

Applications

Money and Compound Interest

We have the formula for compound interest

A = P(1 + r/n)^nt

where A corresponds to the amount in the account after t years in a bank that gives an annual interest rate r compounded n times per year.

Example

Suppose we have $2,000 to put into a savings account at a 4% interest rate compounded monthly. How much will be in the account after 2 years?

We have

P = 2,000, r = .04, n = 12 and t = 2

We want A.

A = 2000(1 + .04/12)¹²⁽²⁾ = $2,166.29.

Continuous Interest

For continuously compounded interest, we have the formula:

A = Pe^rt

Inflation Example

With an 8% rate of inflation in the health industry, how much will health insurance cost in 45 years if currently I pay $200 per month?

Solution

We have

r = .08, P = 200, and t = 45

So that

A = 200e^(.08)(45) = $7319 per month!

Back to the Exponentials and Logarithms Site

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