Download Exponential Functions - MATH 160, Precalculus

Exponential Functions
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Exponential Functions
Objectives
In this lesson we will learn to:
recognize and evaluate exponential functions with base a,
graph exponential functions and use the One-to-One
Property,
recognize and evaluate exponential functions with base e,
use exponential functions to model and solve real-world
problems.
J. Robert Buchanan
Exponential Functions
Exponential Functions
Definition
The exponential function f with base a is denoted by
f (x) = ax
where a > 0, a 6= 1, and x is any real number.
J. Robert Buchanan
Exponential Functions
Exponential Functions
Definition
The exponential function f with base a is denoted by
f (x) = ax
where a > 0, a 6= 1, and x is any real number.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
31.2 =
1.23 =
2−2.7 =
0.80.2 =
1.0112.7 =
J. Robert Buchanan
Exponential Functions
Exponential Functions
Definition
The exponential function f with base a is denoted by
f (x) = ax
where a > 0, a 6= 1, and x is any real number.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
31.2 = 3.73719
1.23 =
2−2.7 =
0.80.2 =
1.0112.7 =
J. Robert Buchanan
Exponential Functions
Exponential Functions
Definition
The exponential function f with base a is denoted by
f (x) = ax
where a > 0, a 6= 1, and x is any real number.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
31.2 = 3.73719
1.23 = 1.728
2−2.7 = 0.153893
0.80.2 = 0.956352
1.0112.7 = 1.1347
J. Robert Buchanan
Exponential Functions
Graphs, a > 1
When a > 1 the graph of an exponential function resembles the
following.
x
−4
−3
−2
−1
0
1
2
3
4
2x
y
1
16
1
8
1
4
1
2
1
2
4
8
16
15
10
5
-4
J. Robert Buchanan
-2
Exponential Functions
2
4
x
Effect of the Base
If we increase the value of the base a, the graph becomes
steeper.
y
30
25
20
15
10
5
-4
-2
J. Robert Buchanan
0
2
Exponential Functions
4
x
Graphs, 0 < a < 1
When 0 < a < 1 the graph of an exponential function
resembles the following.
x
1
y
x
3
35
−4
81
30
27
−3
25
9
−2
20
−1
3
15
0
1
1
10
1
3
1
5
2
9
1
3
27
-4
-2
2
1
4
81
J. Robert Buchanan
Exponential Functions
4
x
Asymptote
Regardless of the base a, the function f (x) = ax approaches,
but never touches the line y = 0 (the x-axis).
In situations like this we may say,
y = 0 is an asymptote of the graph of f (x) = ax , or
y = 0 is a horizontal asymptote of the graph of f (x) = ax ,
or
the graph of f (x) = ax approaches the line y = 0
asymptotically.
J. Robert Buchanan
Exponential Functions
General Concepts of Exponential Functions
For a > 1:
For 0 < a < 1:
ax > 0
ax > 0
ax increases on (−∞, ∞)
and is called an
exponential growth
function.
ax decreases on (−∞, ∞)
and is called an
exponential decay
function.
a0 = 1, so the point (0, 1)
is on the graph.
a0 = 1, so the point (0, 1)
is on the graph.
ax approaches the x-axis
for negative values of x
(The x-axis is a horizontal
asymptote).
ax approaches the x-axis
for positive values of x
(The x-axis is a horizontal
asymptote).
J. Robert Buchanan
Exponential Functions
One-to-One Property
Since exponential functions are either always increasing or
always decreasing, they all pass the Horizontal Line Test and
are therefore one-to-one functions.
One-to-One Property
For a > 0 and a 6= 1, ax = ay if and only if x = y .
J. Robert Buchanan
Exponential Functions
One-to-One Property
Since exponential functions are either always increasing or
always decreasing, they all pass the Horizontal Line Test and
are therefore one-to-one functions.
One-to-One Property
For a > 0 and a 6= 1, ax = ay if and only if x = y .
Example
Use the One-to-One Property to solve the following equation.
2x−3 = 16
J. Robert Buchanan
Exponential Functions
One-to-One Property
Since exponential functions are either always increasing or
always decreasing, they all pass the Horizontal Line Test and
are therefore one-to-one functions.
One-to-One Property
For a > 0 and a 6= 1, ax = ay if and only if x = y .
Example
Use the One-to-One Property to solve the following equation.
2x−3 = 16
2x−3 = 24
J. Robert Buchanan
Exponential Functions
One-to-One Property
Since exponential functions are either always increasing or
always decreasing, they all pass the Horizontal Line Test and
are therefore one-to-one functions.
One-to-One Property
For a > 0 and a 6= 1, ax = ay if and only if x = y .
Example
Use the One-to-One Property to solve the following equation.
2x−3 = 16
2x−3 = 24
x −3 = 4
x
J. Robert Buchanan
= 7
Exponential Functions
Natural Base e
In many applications (particularly to the physical and social
sciences) we will prefer to use the base
e ≈ 2.718281828 . . . .
This is called the natural base and the function f (x) = ex is
called the natural exponential function.
J. Robert Buchanan
Exponential Functions
Natural Base e
In many applications (particularly to the physical and social
sciences) we will prefer to use the base
e ≈ 2.718281828 . . . .
This is called the natural base and the function f (x) = ex is
called the natural exponential function.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
e1.2 =
e3 =
e−2.7 =
e0.2 =
eπ =
J. Robert Buchanan
Exponential Functions
Natural Base e
In many applications (particularly to the physical and social
sciences) we will prefer to use the base
e ≈ 2.718281828 . . . .
This is called the natural base and the function f (x) = ex is
called the natural exponential function.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
e1.2 = 3.32012
e3 =
e−2.7 =
e0.2 =
eπ =
J. Robert Buchanan
Exponential Functions
Natural Base e
In many applications (particularly to the physical and social
sciences) we will prefer to use the base
e ≈ 2.718281828 . . . .
This is called the natural base and the function f (x) = ex is
called the natural exponential function.
Example
Use a calculator if necessary to evaluate each of the following
expressions.
e1.2 = 3.32012
e3 = 20.0855
e−2.7 = 0.0672055
e0.2 = 1.2214
eπ = 23.1407
J. Robert Buchanan
Exponential Functions
Application: Compound Interest
Definition
After t years, the balance A in an account with principal P and
annual interest rate r (expressed as a decimal) is given by the
following formulas.
r n t
1
For n compounding periods per year: A = P 1 +
n
rt
2
For continuous compounding: A = Pe
J. Robert Buchanan
Exponential Functions
Example
Determine the balance A at the end of 20 years if $1500 is
invested at 6.5% interest and the interest is compounded
quarterly
monthly
weekly
continuously
J. Robert Buchanan
Exponential Functions
Example
Determine the balance A at the end of 20 years if $1500 is
invested at 6.5% interest and the interest is compounded
quarterly
0.065 (4)(20)
≈ 5446.73
A = 1500 1 +
4
monthly
weekly
continuously
J. Robert Buchanan
Exponential Functions
Example
Determine the balance A at the end of 20 years if $1500 is
invested at 6.5% interest and the interest is compounded
quarterly
0.065 (4)(20)
≈ 5446.73
A = 1500 1 +
4
monthly
0.065 (12)(20)
A = 1500 1 +
≈ 5484.67
12
weekly
continuously
J. Robert Buchanan
Exponential Functions
Example
Determine the balance A at the end of 20 years if $1500 is
invested at 6.5% interest and the interest is compounded
quarterly
0.065 (4)(20)
≈ 5446.73
A = 1500 1 +
4
monthly
0.065 (12)(20)
A = 1500 1 +
≈ 5484.67
12
weekly
0.065 (52)(20)
A = 1500 1 +
≈ 5499.48
52
continuously
J. Robert Buchanan
Exponential Functions
Example
Determine the balance A at the end of 20 years if $1500 is
invested at 6.5% interest and the interest is compounded
quarterly
0.065 (4)(20)
≈ 5446.73
A = 1500 1 +
4
monthly
0.065 (12)(20)
A = 1500 1 +
≈ 5484.67
12
weekly
0.065 (52)(20)
A = 1500 1 +
≈ 5499.48
52
continuously
A = 1500e(0.065)(20) ≈ 5503.93
J. Robert Buchanan
Exponential Functions
Application: Exponential Growth
The number of fruit flies in an experimental population after t
hours is given by Q(t) = 20e0.03t for t ≥ 0.
1
Find the initial number of fruit flies in the population.
2
How large is the population of fruit flies after 72 hours?
J. Robert Buchanan
Exponential Functions
Application: Exponential Growth
The number of fruit flies in an experimental population after t
hours is given by Q(t) = 20e0.03t for t ≥ 0.
1
Find the initial number of fruit flies in the population.
Q(0) = 20e(0.03)(0) = 20
2
How large is the population of fruit flies after 72 hours?
J. Robert Buchanan
Exponential Functions
Application: Exponential Growth
The number of fruit flies in an experimental population after t
hours is given by Q(t) = 20e0.03t for t ≥ 0.
1
Find the initial number of fruit flies in the population.
Q(0) = 20e(0.03)(0) = 20
2
How large is the population of fruit flies after 72 hours?
Q(72) = 20e(0.03)(72) ≈ 173
J. Robert Buchanan
Exponential Functions
Application: Radioactive Decay
Let Q represent the mass of carbon-14 (14 C) in grams. The
quantity of carbon-14 present after t years is given by
t/5715
1
Q(t) = 10
2
1
Find the initial quantity of carbon-14 present.
2
Determine the quantity present after 2500 years.
J. Robert Buchanan
Exponential Functions
Application: Radioactive Decay
Let Q represent the mass of carbon-14 (14 C) in grams. The
quantity of carbon-14 present after t years is given by
t/5715
1
Q(t) = 10
2
1
Find the initial quantity of carbon-14 present.
0/5715
1
= 10
Q(0) = 10
2
2
grams
Determine the quantity present after 2500 years.
J. Robert Buchanan
Exponential Functions
Application: Radioactive Decay
Let Q represent the mass of carbon-14 (14 C) in grams. The
quantity of carbon-14 present after t years is given by
t/5715
1
Q(t) = 10
2
1
Find the initial quantity of carbon-14 present.
0/5715
1
= 10
Q(0) = 10
2
2
grams
Determine the quantity present after 2500 years.
2500/5715
1
Q(2500) = 10
≈ 7.38441
2
J. Robert Buchanan
Exponential Functions
grams
Half-Life
QHtL
10
8
6
4
2
0
2000
4000
6000
8000
t/5715
1
Q(t) = 10
2
J. Robert Buchanan
Exponential Functions
10 000
t
Homework
Read Section 3.1.
Exercises: 1, 5, 9, 13, . . . , 69, 73
J. Robert Buchanan
Exponential Functions