Post on 11-Jan-2016
CHAPTER 1:PREREQUISITES FOR CALCULUS
SECTION 1.3:EXPONENTIAL FUNCTIONS
AP CALCULUS AB
Exponential GrowthExponential DecayApplicationsThe Number e…and why
Exponential functions model many growth patterns.
What you’ll learn about…
Exponential Function
Let be a positive real number other than 1. The function
( )
is the .
x
a
f x a
a
=
exponential function with base
The domain of f(x) = ax is (-, ) and the range is (0,).
Compound interest investment and population growth are examples of exponential growth.
Exponential Growth
If 1 the graph of looks like the graph
of 2 in Figure 1.22ax
a f
y=
Exponential Growth
If 0 1 the graph of looks like the graph
of 2 in Figure 1.22b.x
a f
y -=
Section 1.3 – Exponential Functions
Example: Graph the function State its domain and range.
3 2 3xf x
Section 1.3 – Exponential Functions
You try: Graph each function State its domain and range.1. 3 6 2. 2 4x xy y
Rules for Exponents
( )
( ) ( )
If 0 and 0, the following hold for all real numbers and .
1. 4.
2. 5.
3.
xx y x y x x
xx xx y
y x
y xx y xy
a b x y
a a a a b ab
a a aa
ba b
a a a
+
-
> >
× = × =
æö÷ç= =÷ç ÷çè ø
= =
b
a x
Rules for Exponents
Rules for Exponents If a > 0 and b > 0, then the following hold for all real numbers
x and y.Rule Example
1.
2.
3.
4.
5.
yxyx aaa 6642 9 333 yx
y
x
aa
a 34
7
xx
x
xyxyyx aaa 842 33
xxx abba
x
xx
b
a
b
a
xxx 632
2733
3
3
33xxx
Half-life
Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.
Note: Carbon-14 half-life is about 5730 years.
Half-life
The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation. In the following equation, n represents the half-life.
nx
ky
2
1
Section 1.3 – Exponential Functions
Example: Suppose the half-life of a certain radioactive substance is 12 days and that there are 8 grams present initially. When will there be only 1.5 grams of the substance remaining?
(Hint: Solve graphically)
Section 1.3 – Exponential Functions
You try: The half-life of a radioactive substance is 20 days. The number of grams present initially is 10 grams. Determine when 4 grams of the substance will remain.
Exponential Growth and Exponential Decay
The function , 0, is a model for
if 1, and a model for if 0 1.
xy k a k
a a
exponential growth
exponential decay
= × >
> < <
Zeros of Exponential Functions
To find the zeros of an exponential function using a graphing calculator (TI-83 or 84):
1. Enter the equation in y1.2. Graph in the appropriate window.3. Use the following keystrokes:
CALC (2nd TRACE)ZEROWhen it says “Left Bound?”, go just left of
the x-intercept and hit ENTER.When it says “Right Bound?”, go just right
of the x-intercept and hit ENTER.When it says “Guess?”, go to approximately
the x-intercept and hit ENTER.It will print out
ZEROx = ________ y = ________
The zero is the x-value.
Example Exponential Functions
( )Use a grapher to find the zero's of 4 3.xf x = -
( ) 4 3xf x = -
[-5, 5], [-10,10]
Section 1.3 – Exponential Functions
Example: Find the zeros of graphically.
7 1.25xf x
Section 1.3 – Exponential Functions
You try: Find the zeros of each functiongraphically.
1.
2. 4 1.25xf x
2 1.20xf x
The Number e
Many natural, physical and economic phenomena are best modeled
by an exponential function whose base is the famous number , which is
2.718281828 to nine decimal places.
We can define to be the numbe
e
e ( ) 1r that the function 1
approaches as approaches infinity.
x
f xx
x
æ ö÷ç= + ÷ç ÷çè ø
The Number e
The exponential functions and are frequently used as models
of exponential growth or decay.
Interest compounded continuously uses the model , where is the
initial investment, is t
x x
r t
y e y e
y P e P
r
-= =
= ×
he interest rate as a decimal and is the time in years.t
Example The Number e
[0,100] by [0,120] in 10’s
Remember
Compounding Formulas:1. Simple Interest:
2. Compounded n times per year:
3. Compounded continuously:
trAtA 10
nt
n
rAtA
10
percent. on the form decimal theis where,0 rePtP rt