Transcript Introduction to Algebra

Exponential Functions
Exponential
Growth and Decay
Functions
Exponential Functions

Growth Rates

Linear Growth
cConsider the arithmetic sequence:
3, 5, 7, 9, 11, 13, ...
a1 = 3
a2 = a1 + 2 = 3 + 2
a3 = a2 + 2 = 3 + 2 + 2
a4 = a3 + 2 = 3 + 2 + 2 + 2
an = a1 + (n – 1)2 = 3 + 2 + 2 + … + 2
n–1 Terms
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Exponential Functions
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Exponential Functions

Growth Rates

Linear Growth

an = a1 + (n – 1)2 = 3 + 2 + 2 + … + 2
n–1 Terms


Define function f(n) by:
f(n) = an = a1 + 2(n – 1)
Let k = n – 1 and define g(k) as
g(k) = a1 + 2k = 3 + 2k = 2k + 3
A Linear
Function
Question: How do we know this is linear ?
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Exponential Functions
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Exponential Functions

Growth Rates

Exponential Growth
 Consider the geometric sequence:
3, 6, 12, 24, 48, 96, ...




a1 = 3
a2 = a1 • 2 = 3 • 21
a3 = a2 • 2 = a1 • 2 • 2 = 3 • 22
an = an–1 • 2 • … • 2 = 3 • 2n–1
n–1 Factors
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Exponential Functions
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Exponential Functions

Growth Rates

Exponential Growth
n–1
 a = a
•
2
•
…
•
2
=
3
•
2
n
n–1
n–1 Factors

Define function f(n) :
f(n) = an = a1 r n – 1 = 3 •2 n – 1
k = n–1
Let k = n – 1 and define g(k) = 3 • 2k
g(k) = a1 • rk = 3 • 2k
An Exponential Function
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Exponential Functions
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Exponential & Power Functions

What’s the difference ?

For any real number x , and rational
number a, we write the ath power of x
as:
a
x
Base x

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Exponent a
Function f(x) = x a is called a power function
Exponential Functions
6
Exponential & Power Functions

What’s the difference ?



Function f(x) = x a is a power function
For any real numbers x and a , with a ≠ 1
and a > 0 , the function f(x) = a x is called
an exponential function
The general form, with C > 0, is :
f(x) = C a x
Constant
Coefficient
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Base
Exponential Functions
Exponent
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Function Comparisons
y
f(x) = 2x
f = { (x, 2x)  x  R }
x –3 –2 –1 0 1 2 3 10
2x –6 –4 –2 0 2 4 6 20
x
Linear Function
y
f(x) = x2
x –3 –2 –1 0 1 2 3 10
x2 9 4 1 0 1 4 9 100
x
f = { (x, x2)  x  R }
Power Function
Question: What is f(5) ? ... and f(20) ?
Which function grows faster as x
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Exponential Functions

?
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Function Comparisons
y
x –3 –2 –1 0 1 2 3 10
x2 9 4 1 0 1 4 9 100
f(x) = x2
f = { (x, x2)  x  R }
x
y
Power Function
f(x) = 2x
x –3 –2 –1 0 1 2 3 10
2x ⅛ ¼ ½ 1 2 4 8 1024
x
f = { (x, 2x)  x  R }
Exponential Function
Question: What is f(5) ? ... and f(20) ?
Which function grows faster as x
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Exponential Functions

?
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Exponential Functions

Increasing and Decreasing

Exponential growth :
f(x) = Ca x
y
f(x) = C2 x
for a > 1 and C > 0


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(0, C) ●
Example :
f(x) = C2x
x
Domain = R
Range = { x  x > 0 }
Exponential decay :
Exponential Functions
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Exponential Functions

Increasing and Decreasing


Exponential growth :
f(x) = Ca x
Exponential decay :
g(x) = f(–x) = Ca –x
for a > 1 , say a = 2
y
h(x)
f(x) = C2 x
(0, C) ● g(x) = f(–x) = C2
–x
x
... a reflection of f(x)
1
Domain = R
Range = { x  x > 0 }
OR , for 0 < b < 1 and b = a
h(x) = Cbx = C
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1 x
2
( )
Exponential Functions
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Exponential Functions

Increasing and Decreasing


Exponential growth :
f(x) = Ca x
Exponential decay :
g(x) = f(–x) = Ca –x
y
f(x) = C2 x
(0, C) ● g(x) = f(–x) = C2
Questions:
Intercepts ? One
Asymptotes ? One
Growth factor a ? a > 1
Decay factor a ? 0 < a < 1
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Exponential Functions
–x
x
Domain = R
Range = { x  x > 0 }
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Exponential Functions

Increasing and Decreasing


Exponential growth :
f(x) = Ca x
Exponential decay :
g(x) = f(–x) = Ca –x
y
f(x) = C2 x
(0, C) ● g(x) = f(–x) = C2
As ordered pairs,
with C = 1 ,
f = { (x, 2x)  x  R }
and
g = { (x, 2–x )  x  R }
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Exponential Functions
–x
x
Domain = R
Range = { x  x > 0 }
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Exponential Functions

Increasing and Decreasing


Exponential growth :
f(x) = Ca x
Exponential decay :
g(x) = f(–x) = Ca –x
In tabular form,
with C = 1 ,
y
f(x) = C2 x
(0, C) ● g(x) = f(–x) = C2
x
x –4 –3 –2 –1 0 1 2 3 4
2x ⅛16 ⅛ ¼ ½ 1 2 4 8 16
2–x 16 8 4 2 1 ½ ¼ ⅛ ⅛16
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–x
Exponential Functions
Domain = R
Range = { x  x > 0 }
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Exponential Function Basics

Let f(x) = ax with a > 0 , a ≠ 1

f(0) = 1

Domain-of-f = R = ( – ∞, ∞)

R
Range-of-f = { y | x > 0 } = ( 0 , ∞)

Graph is increasing for a > 1 and
decreasing for a < 1
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Exponential Functions
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Exponential Function Basics

Let f(x) = ax with a > 0 , a ≠ 1

f is 1–1

For a > b > 1 :
ax > bx for x > 0
and ax < bx for x < 0
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WHY ?

Graphs of ax and bx intersect at (0, 1)

If ax = ay then x = y
Exponential Functions
WHY ?
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Natural Exponential Function
Compute the first few terms of the sequence an =
n
an
1
2
3
4
5
6
10
100
200
400
2000
10000
100000
1000000
2.000000000
2.250000000
2.370370370
2.441406250
2.488320000
2.521626372
2.593742460
2.704813829
2.711517123
2.714891744
2.717602569
2.717942121
2.718268237
2.718280469
1
1+ n
(
)
n
Question:
Does an approach a value as n
∞?
In fact,
an
2.7182 81828 45904 52353 60287 ....
We call this number e
e is irrational (in fact transcendental)
and is the base for natural exponential
functions ... and natural logarithms
Natural exponential functions are of form
f(x) = ex
?
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Exponential Functions
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Natural Exponential Function
Graph of f(x) =
x
ex

ex
5
1
2.7182 81828
2
7.3890 56098
3
20.0855 36923
4
54.5981 50033
5
148.4131 59105
6
403.4287 93492
7
1096.6331 58428
8
2980.9579 87041
9
8103.0839 27575
10
22026.4657 94806
11
59874.1417 15197
12 162754.7914 19003
13 442413.3920 08920
14 1202604.2841 64776
?
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ex
ex

1200
f(x) = ex
4

3

1000
2
800

600
–1

1


x
0
1
2
3

400
200





1
2
3
Exponential Functions

4
5
6
7
x
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Continuous Compounding
We have shown that
(
1 + n1
Thus
n
)
1 nx =
1+ n
x
1 n
1+ n
1
and n
)
((
as n
∞
(
∞
e as n
1
and n
))
e
0
x
0
Note: For any positive constant r,
(
r
1+ n
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)
n
r
e
as nr
Exponential Functions
∞
r
and n
0
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Continuous Compounding
Note:
(
1 + nr
)
n
r
e
∞
as nr
r
and n
0
Recall: amount P compounded n times per
year at annual interest rate r for t years is
A = P(1 + r/n)nt
= P(1 + r/n)(n/r)rt
= P((1 + r/n)(n/r))rt
Pert
e
What does this mean ?
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Exponential Functions
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Continuous Compounding
A = P(1 + r/n)nt
Pert
As the number of compounding periods per
year (n) increases without bound, periodic
compounding approaches continuous
compounding
Thus amount P compounded continuously
for t years at annualized interest rate r yields
amount A given by
A = Pert
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Exponential Functions
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Exponential Equations

Solve

1. 25x = 125
(52)x = 53
52x = 53
2x = 3 WHY ?
x = 3/2
Solution set is
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3
2
{ }
Exponential Functions
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Think about it !
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Exponential Functions
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