Exponential Functions
Exponential Functions are functions of the form y = abx.
(Notice: If the domain is
limited to natural numbers (1,2,3,...), the function
becomes a geometric sequence.)
Graph y = 2x in your calculator.
Notice
it has the following characteristics.
- Domain: All Real Numbers
- Range: y > 0
- Asymptote: y = 0 (the x-axis)
- It is a 1-1 function
- It passes through (0,1)
All exponential functions have the same properties as you can see if
you
graph others in
your calculator.
Example:
Find the exponential equation for which h(0) = 5 and h(1)
= 15.
- The two points the function passes through are (0,5)
and (1,15).
- An exponential function has the form y = abx.
- Substitute the two points into the equation:
5 = ab0
15 = ab1 - Solve the first one for a:
5 = a(1)
5 = a - Substitute the a
value obtained into the second
equation.
15 = 5b - Solve for b:
3 = b - So y = 5(3)x
Another Example:
The value of a $24,000 machine decreases 9%
each year. What will the machine be worth
in 3 years? In how many years will the value of
the machine be about half its value when new?
Use A = A0(1 + r)n where A
is the amount after n years, A0is
the initial amount, r is the
annual interest rate as a decimal, and n is
the number of years.
A = 24,000 (1 - .09)3 (subtraction
because the amount is decreasing)
= $18085.70
Rule of 72:
The rule of 72 is a way of figuring how long it takes
money
to double in a savings account. It says the doubling time is
equal
to 72 divided by the annual interest rate.
If a table is built of the doubling times at different interest
rates,
using 2A = A(1 +r)n, the following is
obtained.
| rate |
4% |
6% |
8% |
doubling
time |
18 |
12 |
9 |
If each rate is divided into 72, the doubling time
is obtained. It is not perfectly accurate every time, but it is
close
enough with which to estimate.
Half-Life Problems:
Use the formula: A = A0(1/2)t/h
Example: The half-life of radium
is
1600
years. If 1 kg is present now, how much
will be present after
3200 years?
A = 1(1/2)3200/1600 = (1/2)2
= 1/4 kg
Natural Exponential Function:
What is e?
e
= lim (1 + 1/n)n
n®¥
Figuring Amount in Savings Account with Compound Interest:
Let r = annual percentage interest
n = number of compounding
periods
in a year
A0 = amount deposited
Find the amount of money in a savings account after n
years.
- A1 = A0 +(r/4) A0
=
A0 (1 + r/4 )
- A2 = A0 (1 + r/4 )
+
(r/4) A0(1
+ r/4 ) = A0 (1 + r/4 )(1
+
r/4 ) + A0(1 + r/4)2
- Similarly, A3 = A0(1 +
r/4)3
- and An = A0(1 + r/4)n
Formula for t years:
At = A0(1 + r/n)nt
Derive the formula for compounding continuously:
Start with the formula for t years and let n approach
oo.
- At = A0(1 + r/n)nt
- Let m = n/r
- Then A = Ao(1 + 1/m)mrt
= Ao[(1 + 1/m)m]rt - As m
approaches , A approaches Aoert
- Thus A = Aoert for
continuous
compounding
Problems
Go to Exponential
& Logarithmic Functions Lesson Page
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