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Nonlinear ModelsChapter2
• Quadratic Functions and Models
• Exponential Functions and Models
• Logarithmic Functions and Models
• Logistic Functions and Models
Lectures 2 & 3
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Quadratic FunctionQuadratic Function
2( ) 0f x ax bx c a
A quadratic function of the variable x is a function that can be written in the form
Ex.
a, b, and c are fixed numbers
2( ) 12 3 1f x x x
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Quadratic FunctionQuadratic Function
2( ) 0f x ax bx c a
Every quadratic function has a parabola as its graph.
a > 0a < 0
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Features of a ParabolaVertex:
x – intercepts
y – intercept
symmetry
,2 2
b bx y f
a a
2 0ax bx c 2
bx
a
y c
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Sketch of a Parabola
Vertex:
x – intercepts
y – intercept
2; 92
bx y
a
5y
2( ) 4 5f x x x Ex.
2 4 5 0x x 5,1x
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Application
Ex. For the demand equation below, express the total revenue R as a function of the price p per item and determine the price that maximizes total revenue.
3 600q p
( ) 3 600R p pq p p 23 600p p
Maximum is at the vertex, p = $100
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Exponential FunctionExponential Function
( ) 0xf x Ab b
An exponential function with base b and exponent x is defined by
Ex. ( ) 5 3xf x
where A and b are constants.
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Laws of ExponentsLaws of ExponentsLaw Example
1. x y x yb b b
2.x
x yy
bb
b
4.x x xab a b
3.yx xyb b
5.x x
x
a a
b b
1/ 2 5 / 2 6 / 2 32 2 2 2 8 12
12 3 93
55 5
5
61/ 3 6 / 3 2 18 8 8
64
3 3 3 32 2 8m m m 1/ 3 1/ 3
1/ 3
8 8 2
27 327
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Graphing Exponential FunctionsGraphing Exponential Functions
Ex. ( ) 3xf x
(0,1)
( )y f x
0 1
1 3
2 9
11 3
x y
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Finding the Exponential Curve Through Two Points
Ex. Find an equation of the exponential curve that passes through (1,10) and (3,40).
( ) xf x Ab
110 Ab340 Ab
340
10
Ab
Ab
24 b2b
Plugging in we get A = 5
( ) 5 2 xf x
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ExampleExampleEx. A certain bacteria culture grows according to the following exponential growth model. The bacteria numbered 20 originally, find the number of bacteria present after 6 hours.
0.4479( ) 20 4 tQ t
0.4479(6)(6) 20 4 829.86Q
So about 830 bacteria
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Compound InterestCompound Interest
( ) 1mt
rA t P
m
A = the future value
P = Present valuer = Annual interest ratem = Number of times/year interest is compoundedt = Number of years
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Compound InterestCompound Interest
Find the accumulated amount of money after 5 years if $4300 is invested at 6% per year and interest is reinvested each month
1mt
rA P
m
12(5).06
4300 112
A
= $5800.06
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The Number e
e is an irrational constant.
2.718281828459045...e
If $1 is invested for 1 year at 100% interest compounded continuously (m gets very large) then A converges to e:
11
m
A em
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Continuous Compound InterestContinuous Compound Interest
rtA Pe
A = Accumulated amount P = Present valuer = Annual interest ratet = Number of years
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Ex. Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.
rtA Pe0.12(25)7500A e
$150,641.53
Continuous Compound InterestContinuous Compound Interest
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LogarithmsLogarithms
log if and only if 0yby x x b x
The base b logarithm of x is the power to which we need to raise b in order to get x.
Ex. 3
7
1/ 3
5
log 81 4
log 1 0
log 9 2
log 5 1
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Logarithms on a Calculator
Common Logarithm10log log
ln loge
x x
x x
Natural Logarithm
Abbreviations
log 4 0.60206
ln 26 3.2581
Base 10
Base e
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Change-of-Base Formula
To compute logarithms other than common and natural logarithms we can use:
log lnlog
log lnba a
ab b
9log15
log 15 1.232487log9
Ex.
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Logarithmic Function GraphsLogarithmic Function Graphs
Ex. 3( ) logf x x
(1,0)
3logy x1/ 3logy x
1/ 3( ) logf x x
(1,0)
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Properites of Logarithms
1. log log log
2. log log log
3. log log
4. log 1 0
5. log 1
b b b
b b b
nb b
b
b
mn m n
mm n
n
m n m
b
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Ex. How long will it take a $800 investment to be worth $1000 if it is continuously compounded at 7% per year?
0.071000 800 te
3.187765t
0.075
4te
5ln 0.07
4t
Apply ln to both sides
Application
About 3.2 years
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Logarithmic FunctionLogarithmic Function
( ) log bf x x C A logarithmic function has the form
Also:( ) ln f x A x C
Ex. ( ) 4.6 ln 8f x x
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ExampleExampleSuppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 0.3( ) 200 150 tT t e
1. Find the temperature after 5 minutes.0.3(5)(5) 200 150 166.5T e
2. Find the time it takes to reach 190°.0.3190 200 150 te
0.31/15 te ln 1/15
9 min.0.3
t
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Logistic FunctionLogistic Function
where A, N, b are constants.
A logistic function is a function that may be expressed in the form:
( ) 0, 11 x
Nf x b b
Ab
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Logistic FunctionLogistic Function
( ) 0, 11 x
Nf x b b
Ab
N N
b >1 0 < b <1
N is called the limiting value
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Logistic Function for Small Logistic Function for Small xx
Thus it grows approximately exponentially with base b.
For small values of x we have:
11x
x
N Nb
AAb
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ModelingModelingEx. A small school district has 2400 people. Initially 10 people have heard a particular rumor and the number who have heard it is increasing at 50%/day. It is anticipated that eventually all 2400 people will hear the rumor. Find a logistic model for the number of people who have heard the rumor after t days.
10(1 ) 2400A Using (0,10):
tAb
tP
1
2400
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For small value of t: in 1 day 15 people will know
so b = 1.503
A = 239
tb
tP
2391
2400
12391
240015
b
t
tP
50312391
2400
.