5.3.1 Exponential Functions and Models I
Consider:
f(x) = 2
x
Do you see anything different about this function?
Numerical (table form):
x
-2
-1
f(x)
0
1
2
Graph it:
In general, an exponential function is defined by
f(x) = Cax
a > 0, a  1, C > 0
a is referred to as the base of the exponential
C can be referred to as the initial value when x represents
time (i.e., when time = 0)
5.3.1-1
For f(x) = 2x:
x
f(x)
-2
-1
0
1
2
1/4
1/2
1
2
4
¼x2 ½x2 1x2
2x2 4x2
Note that for inputs one unit apart
 each output = previous output x base
 this is characteristic of exponential functions
Exponential growth and decay
x
f(x) = 2
a > 1  exponential growth
3
8
f(x) = (1/2)x
a < 1  exponential decay
Note that
x
1
1
f(x) = 2-x = x =   = (1/2)x
2
2
(negative exponent and a > 1) also represents decay
5.3.1-2
Compound interest - exponential growth




you place $100 in the bank
it earns 10% per year compound interest
how much do you have after 1 year?
100 + .10(100) = 100(1 + .10) = 100(1.10) = $110.00






growth multiplier
growth multiplier: what you multiply the old amount
by to get the new amount
notice: the .10 part of 1.10 = the interest rate!
end of second year = old amount x growth multiplier
= 110(1.10) = $121.00 (NOT $120.00 !!)
Compound interest refers to interest (the $10) going back
into the investment to earn subsequent interest on itself
COMPUTING THE AMOUNT AFTER ANY NUMBER OF YEARS
end calculation: (old amount)(growth factor) = new
year
amount
1
100(1.10)
$110.00
2
[100(1.10)](1.10) = 100(1.10)2
$121.00
2
3
3
[100(1.10) ](1.10) = 100(1.10)
.
.
.
10 100(1.10)10
259.37
n
100(1 + .10)n
if A0 = initial principal, An = amount after n years, r = rate,
An = A0(1 + r)n
5.3.1-3
COMPOUNDING PERIODS
 money doesn’t have to be compounded annually
 can be for shorter or longer periods
 could “add in” the interest every
 period of 6 months  compounded semiannually
 period of 3 months 
"
quarterly
 period of one day 
"
daily
let m = number of periods in a year, r = yearly interest rate,
n = number of years
Suppose we invest $100 compounded quarterly at a yearly
rate of 12% for 20 years:
 since the yearly interest rate is 12%, the rate per period
is 12%  4 = 3% per period
 since the the number of years is 20, the number of
periods is 20  4 = 80 periods
 now we can use our formula: A = 100(1 + .12/4)(20)(4)
Summarizing: we let
 exponent = the number of periods (not years) =
 rate = the rate per period (not per year) = r/m
mn
5.3.1-4
A GENERAL FORMULA FOR COMPOUND INTEREST
Make the following definitions:
A 0:
principal amount invested
m:
number of periods in a year
n:
number of years principal is invested
r:
interest rate per year
A n:
total amount in account at end of n years
Then:
r

An = A0 1  
 m
mn
An example:
Hillary invests $10,000 in Arkansas razorback pig futures
projected to accumulate earnings at 20% per year compounded
quarterly. How much will her investment be worth in 10 years?
.20 (4)(10)
A10 = 10000(1+
)
= 10000(1.05)40
4
= 10000(7.0400) = $70,400
Her investment today of $10,000 should accumulate to a
walloping $70,400 in just 10 years time!
Exponential functions with base e
f(x) = ex




e is a real number constant (like ) value = 2.7182818…
frequently seen as the base for exponential functions
called the natural base
it arises “naturally” in many mathematical contexts
5.3.1-5
Compound interest and the number e
 invest $1.00 at 100% interest compounded annually,
how much do you have at the end of 1 year?
A=
 compound semiannually? A = 1(1 + 1.00/2)2 =
 for n periods per year, we have A = (1 + 1/m)m
Using our formula (and our calculator) we can build a table:
Compounding:
annually
semiannually
monthly
daily
hourly
every minute




m=1
m=2
m = 12
m = 365
m = 8760
m = 525600
A
A
1
(1 + 1/1)
(1 + 1/2)2
(1 + 1/12)12
(1 + 1/365)365
(1 + 1/8760)8760
(1 + 1/525600)525600
$2.00
$2.25
$2.61304 . . .
$2.71457 . . .
$2.71826 . . .
$2.71827 . . .
no matter how many compounding periods per year
our account will approach $2.71828 . . .
but never reach it (it’s an asymptote!)
it’s the number e again!!
 e can be thought of as follows:
e = the dollar amount you will have if you invest
$1.00 for 1 year at 100% interest
compounded continuously
Formula for continuous compounding (n = # of years):
An = A0ern
5.3.1-6