Transcript Chapter 1.3

AP CALCULUS AB
CHAPTER 1:
PREREQUISITES FOR CALCULUS
SECTION 1.3:
EXPONENTIAL FUNCTIONS
What you’ll learn about…
Exponential Growth
Exponential Decay
Applications
The Number e
…and why
Exponential functions model many growth patterns.
Exponential Function
Let a be a positive real number other than 1. The function
f ( x) = a x
is the exponential function with base a.
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 a 1 the graph of f looks like the graph
of y= 2 x in Figure 1.22a
Exponential Growth
If 0  a 1 the graph of f looks like the graph
of y = 2- x in Figure 1.22b.
Section 1.3 – Exponential Functions
 Example: Graph the function
State its domain and range.
 
f  x   3 2x  3
Section 1.3 – Exponential Functions
 You try: Graph each function State its domain and
range.
1. y  3x  6
2. y  2 x  4
Rules for Exponents
If a > 0 and b > 0, the following hold for all real numbers x and y.
x
1. a x ×a y = a x + y
4. a x ×b x = (ab)
ax
2. y = a xa
x
x
æ
aö
a
a


5. çç ÷
÷
÷ = b x
çèbbø
3. (a
x
y
x
y
y x
) = (a ) = a xy
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.
a x  a y  a x y
2.
32  34  36
ax
x y

a
ay
x7
3

x
x4
3.
a   a 
4.
a x  b x  ab 
5.
x y
y x
a
x
x
ax
a
   x
b
b
 96
xy
3 
2 4
 38
2 x  3x  6 x
3
x3 x3
 x
   3 
3
27
 3
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.
1
y  k 
 2
x
n
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 y = k ×a x , k > 0, is a model for exponential growth
if a > 1, and a model for exponential decay if 0 < a < 1.
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)
ZERO
When 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
ZERO
x = ________
y = ________
The zero is the x-value.
Example Exponential Functions
Use a grapher to find the zero's of f (x)= 4 x - 3.
f (x)= 4 x - 3
[-5, 5], [-10,10]
Section 1.3 – Exponential Functions
 Example: Find the zeros of
graphically.
f
 x   7  1.25x
Section 1.3 – Exponential Functions
 You try: Find the zeros of each function
graphically.
1.
f  x   2 1.20x
2.
f  x   4  1.25x
The Number e
Many natural, physical and economic phenomena are best modeled
by an exponential function whose base is the famous number e, which is
2.718281828 to nine decimal places.
æ
We can define e to be the number that the function f (x)= çç1 +
çè
approaches as x approaches infinity.
x
ö
1÷
÷
ø
x÷
The Number e
The exponential functions y = e x and y = e- x are frequently used as models
of exponential growth or decay.
Interest compounded continuously uses the model y = P ×e r t , where P is the
initial investment, r is the interest rate as a decimal and t is the time in years.
Example The Number e
[0,100] by [0,120] in 10’s
Remember
Compounding Formulas:
1. Simple Interest:
At   A0  1  r 
t
2.
3.
Compounded n times per year:
 r
At   A0  1  
 n
nt
Compounded continuously:
Pt   P0 e rt , where r is the decimal form on the percent.