6.1 Exponential Functions
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Transcript of 6.1 Exponential Functions
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Transcendental
Functions6.1 Exponential Functions
RA Idoy
MATH17
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• Theorems
• Simple interest
• Compound interest
• The number e
• Exponential Function with baseb
• Natural Exponential Function
• Application
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Theorems
Theorem 1.
If r and s are rational
numbers, then
(i) if b>1, r<s implies br <bs,
(ii)if 0<b<1, r<s implies br>bs
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Theorems
Theorem 2.
If a and b are any positive
numbers, and x and y are any
real numbers, then
)
)
)
x y x y
x
x y y
y x xy
i a a a
aii aa
iii a a
)
)
x x x
x x
x
iv ab a b
a av
b b
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Simple Interest
Definition:
Simple interest is interest
due only on the original amount
that is borrowed. It is given
by the formula
where
P dollars is the investment, i is the
simple interest rate, n is the years
1 A P Pni P ni
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Simple Interest
1 A P Pni P ni
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Simple Interest
Example:
A loan of $500 is made
for a period of 90 daysat a simple interest rate
of 16 percent annually.
Determine the amount tobe repaid at the end of
90 days.
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Simple Interest
Solution:
Let P = 500, i=0.16, n=90/360
Therefore, the amount to be repaid at
the end of 90 days is $520.
1
1500 1 0.16
4
520
A P ni
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Compound Interest
When the interest for each
period is added to the
principal and itself earns
interest, we have compound interest.
1
mt
i A Pm
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Compound Interest
Example:
Suppose that $400 isdeposited into a saving account
that pays 8 percent per yearcompounded semiannually. If nowithdrawals and no additionaldeposits are made, what is theamount on deposit at the end of3 years?
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Compound Interest
Solution: Let P = 400, i=0.08, m=2,
t=3
The amount of deposit at the end of 3
years is therefore $506.13.
2 3
6
1
0.08400 1
2
400 1.04
506.13
mt i
A Pm
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Compound Interest
Let P=1,
i=1, and
t=1.
1
11
mt
m
i A P
m
m
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The number e
Continuous compound interest
2.7182818e
it A Pe
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Exponential Function with
base b
If b>0 and b≠1, then the
exponential function with base
b is the function f defined by
Note:
x f x b
Dom f
Rng f
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Natural Exponential
Function
The natural exponential function
is the function f defined by
Note:
x
f x e
Dom f
Rng f
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Applications
• Exponential Growth
• Exponential Decay
•
Bounded Growth
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Exponential Growth
Usually applied to:
Bacteria growth
Population growth
Given by the function:
0kt f t Be t
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Exponential Growth
Graph: 0kt f t Be t
B
t
f(t)
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Exponential Growth
Example: In a particular bacteria
culture, if f(t) bacteria are
present at t minutes, then
where B is a constant. If there are
1500 bacteria present initially,how many bacteria will be present
after 1 hour?
0.04t f t Be
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Exponential Growth
Solution:
Given: At time
t=0, f(0)=1500. After one hour…
0.04
0.04 0
0
0
1500
1500
t f t Be
f Be
Be
B
0.041500
t f t e
0.04 60
2.4
60 1500
1500
1500 11.023
16535
f e
e
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Exponential Growth
Conclusion:
Therefore, there are 16535
bacteria present in the culture
after 1 hour.
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Exponential Decay
Usually applied to:
Radioactivity
Sales
Given by the function:
0kt f t Be t
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Exponential Decay
Graph: 0kt f t Be t
B
t
f(t)
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Exponential Decay
Example: If V(t) dollars is thevalue of a certain piece ofequipment t years after itspurchase, then
where B is a constant. If the
equipment was purchased for$8000, what will be its value in2 years?
0.20t V t Be
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Exponential Decay
Solution:
Given: At time
t=0, V(0)=8000. After 2 years…
0.20
0.20 0
0
0
8000
8000
t V t Be
V Be
Be
B
0.208000
t V t e
0.20 2
0.4
2 8000
8000
8000 0.670320
5362.56
V e
e
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Exponential Decay
Conclusion:
Therefore, the value of the
equipment in 2 years will be
$5362.56.
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Bounded Growth
Given by the function:
1 0
kt
f t A e t
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Bounded Growth
Graph:
A
t
f(t)
1 0kt f t A e t
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Bounded Growth
Example: A typical worker at a certainfactory can produce f(t) units perday after t days on the job, where
(a)How many units per day can theworker produce after 7 days on thejob?
(b)How many units per day can theworker eventually be expected toproduce?
0.34
50 1t
f t e
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Bounded Growth
Solution for a:
We wish to find
f(7)
Conclusion:
The worker can
produce 45 units
per day after 7days on the job.
0.34 7
2.38
7 50 1
50 1
50 1 0.093
45
f e
e
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Bounded Growth
Bounded growth is also described
by a function defined by
kt
f t A Be
0,t A B