Exponential Functions

• An exponential function is a function of the form the

real constant a is called the base, and the independent variable x may assume any real value.

• The graph of y = 2x is shown below. It is increasing as are all exponential functions with base > 1.

, 1 ,0 ,)( aaaxf x

More about Exponential Functions

• The graph of y = is shown next. It is decreasing as are all exponential functions with 0 < base < 1.

• Since exponential functions are increasing or decreasing, it follows that they are one-to-one. Why?

• By examining the graph we conclude that the range of an exponential function is the set of positive real numbers.

x

2

1

More about Exponential Functions

• The graph of always passes through the points (0, 1) and (1, a).

• The graph of is the reflection about the y-axis of the

graph of

• The graph of has the horizontal asymptote y = 0.

• If there are two exponential functions and a < b, then

xaxf )(

.1

)( xx

aa

xf

xaxf )(

xaxf )(

xx bxgaxf )( and )(0. for )()( and 0for )()( xxgxfxxgxf

Solving Exponential Equations

• If

• Example. Solve 310 = 35x. By the previous bullet points,

• Example. Solve. 27 = (x–1)7. By the previous bullet points,

,1 0,, 1 ,0 bbaa.vuaa vu

.0 assuming , ubaba uu

.2

510

x

x

.3

12

x

x

The Number e

• As the real number m gets larger and larger,

• The limiting value 2.71828... is an irrational number known as e.

• In order to simplify certain formulas, exponential functions are often written with base e.

• For x > 0, 2x < ex < 3x.

2.71828... approaches )1( 1 mm

Compound Interest

• When the money in an account receives compound interest, each interest payment includes interest on the previously accrued interest.

• Example. $100 compounded annually at 10% interest for 3 years, and P dollars compounded annually at r% interest for 3 years

Year Starting Amount

Ending Amount StartingAmount

Ending Amount

1 100 100(1+0.1) = 110 P P(1+r)

2 110 110(1+0.1) = 121 P(1+r) P(1+r)2

3 121 121(1+0.1) = 133.10 P(1+r)2 P(1+r)3

Compound Interest Formula

• If the interest on an account at r% annually is compounded k times per year, the interest rate applied to each accounting period is r/k.

• When k = 2, we say that interest is compounded semiannually, when k = 4, we say that interest is compounded quarterly, and when k = 12, we say that interest is compounded monthly.

• In general, if P dollars are invested at an annual interest rate r (expressed in decimal form) compounded k times annually, then the amount A available at the end of t years is

.)1( ktkrPA

Compound Interest Example• Suppose that $6000 is invested at an annual rate of 8%. What

will be the value of the investment after 3 years if(a) interest is compounded quarterly?

(b) interest is compounded semiannually?

• In which case, (a) or (b), is the total amount of interest greater? Why?

45.7609$)02.01(6000 12 A

91.7591$)04.01(6000 6 A

Continuous Compounding

• Suppose we let the number of compounding periods k increase without bound. (imagine compounding every second, then every millisecond, etc.). The amount of the investment of P dollars after t years approaches a limit:

• When the above situation pertains, we say that we are compounding continuously.

• In general, if P dollars are invested at an annual interest rate r (expressed in decimal form) compounded continuously, then the amount A available at the end of t years is

. where

,)1()1( 1

rk

rtrtmm

ktkr

m

PePPA

.rtPeA

Compound Interest Examples

• Example. Suppose that $6000 is invested at an annual rate of 8%. What will be the value of the investment after 3 years if interest is compounded continuously?

• Note that the amount of the investment after 3 years is greater than it was when compounding was done semiannually or quarterly? Why?

• Example. Suppose that a principal P is to be invested at continuous compound interest of 8% per year to yield $10,000 in 5 years. How much should be invested?

49.7627$)71828.2(6000 )3)(08.0( A

20.67034918.1

1000010000)5)(08.0(

e

P

Exponential Growth Model--World Population

• A model which predicts the quantity Q, which is number or biomass, for a population at time t is the following exponential growth model:

Both

q0 and k are constants specific to the particular population in question, and k is called the growth constant.

• For the world population, k = 0.019 and q0 = 6 billion when t = 0 corresponds to the year 2000. The model is:

In the year 2010, the model predicts a world population of

.0,0,)( 00 kqeqtQ kt

years.in ,6)( 019.0 tetQ t

billion. 7.266)10( )10(019.0 eQ

Exponential Decay Model

• A model which predicts the quantity Q, which is mass, for a particular radioactive element at time t is the following exponential decay model:

Both q0 and k are constants specific to the

particular radioactive sample in question, and k is called the decay constant. We use the term half-life to describe the time it takes for half of the atoms of a radioactive element to break down.

• A radioactive substance has a decay rate of 5% per hour. If 500 grams are present initially, how much remains after 4 hours?

.0,0,)( 00 kqeqtQ kt

grams.37.409500)4(

,500)()4)(05.0(

05.0

eQ

etQ t

Summary of Exponential Functions; We discussed

• Definition of an exponential function and its base

• Fact that exponential functions are increasing or decreasing and therefore they are one-to-one

• Range of an exponential function

• Horizontal asymptote of an exponential function

• Solving exponential equations

• The number e

• The formula for compound interest

• The formula for continuous compounding

• Exponential growth and decay