1. Lesson 1.3: Exponential Functions
AP Calculus
Mrs. Mongold

2. Review of Exponent Rules
ax ∙ ay = ax+y
(ax)y = axy
ax ∙ bx = (ab)x

3. Exponential Function
Let a be a positive real number other than 1. Then f(x) = ax is the exponential function with base a.
Domain of f(x) = ax is (-∞, ∞)
Range of f(x) = ax is (0, ∞)

4. Quick Examples
y = 2x
y = 2-x

5. Exponential Growth y = Pax P = initial amount
a = 1 + rate as a decimal
x = # of times compounded
F(x) = P(1+r)x

6. Example
Find growth of $100 invested in 2009 at an interest rate of 5.5% in 2013 which was compounded annually.
Find amount in 2020

7. Example
Predict the world population in 2020 using an exponential model and the table
Year
Population (millions)
Ratio
2005
4936
2006
5023
2007
5111
2008
5201
2009
5329
2010
5422

8. Exponential Decay
The function y=kax
k>0 and 0<a<1

9. Half-life example
A certain radioactive substance has a half-life of 20 days. If there are 5 grams present initially when will there be only 1 gram remaining?

10. The number e e=
We can define e to be the number that the function f(x) = (1+ 1/x)x approaches as x approaches infinity.
y=ex and y=e-x are frequently used as models for exponential growth and decay.

11. Example
y=Pert is used for interest compounded continually

12. Homework
Pgs 24-25/1-6 all, 7-17 odd, 23, 25, 29, 30, 33