Chapter 2 Nonlinear Models Sections 2'1, 2'2, and 2'3

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Chapter 2Nonlinear ModelsSections 2.1, 2.2,
and 2.3
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Nonlinear Models

• Quadratic Functions and Models
• Exponential Functions and Models
• Logarithmic Functions and Models

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Quadratic Function
A quadratic function of the variable x is a
function that can be written in the form
where a, b, and c are fixed numbers
Example
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Quadratic Function
The graph of a quadratic function is a parabola.
a gt 0
a lt 0
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Vertex, Intercepts, Symmetry
Vertex coordinates are
y intercept is
symmetry
x intercepts are solutions of
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Graph of a Quadratic Function
Example 1 Sketch the graph of
Vertex
y intercept
x intercepts
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Graph of a Quadratic Function
Example 2 Sketch the graph of
Vertex
y intercept
x intercepts
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Graph of a Quadratic Function
Example 3 Sketch the graph of
Vertex
y intercept
x intercepts
no solutions
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Applications
Example 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.
Maximum is at the vertex, p 100
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Applications
Example As the operator of Workout Fever health
Club, you calculate your demand equation to be q
??0.06p 84 where q is the number of members in
the club and p is the annual membership fee you
charge. 1. Your annual operating costs are a
fixed cost of 20,000 per year plus a variable
cost of 20 per member. Find the annual revenue
and profit as functions of the membership price
p. 2. At what price should you set the membership
fee to obtain the maximum revenue? What is the
maximum possible revenue? 3. At what price should
you set the membership fee to obtain the maximum
profit? What is the maximum possible profit? What
is the corresponding revenue?
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Solution
The annual revenue is given by
The annual cost as function of q is given by
The annual cost as function of p is given by
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Solution
Thus the annual profit function is given by
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The graph of the revenue function
is
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The graph of the revenue function
is
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The profit function is
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The profit function is
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Nonlinear Models

• Quadratic Functions and Models
• Exponential Functions and Models
• Logarithmic Functions and Models

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Exponential Functions
An exponential function with (constant) base b
and exponent x is defined by
Notice that the exponent x can be any real number
but the output y bx is always a positive
number. That is,
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Exponential Functions
We will consider the more general exponential
function defined by
where A is an arbitrary but constant real number.
Example
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Graph of Exponential Functionswhen b gt 1
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Graph of Exponential Functionswhen 0 lt b lt 1
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Graph of Exponential Functionswhen b gt 1
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Graphing Exponential Functions
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Graphing Exponential Functions
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Laws of Exponents
Law
Example
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Finding the Exponential Curve Through Two Points
Example Find an exponential curve y ?Abx that
passes through (1,10) and (3,40).
Plugging in b ? 2 we get A ? 5
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Exponential Functions-Examples
A certain bacteria culture grows according to the
following exponential growth model. If the
bacteria numbered 20 originally, find the number
of bacteria present after 6 hours.
Thus, after 6 hours there are about 830 bacteria
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Compound Interest
A the future value
P Present value
r Annual interest rate (in decimal form)
m Number of times/year interest is compounded
t Number of years
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Compound Interest
Find the accumulated amount of money after 5
years if 4300 is invested at 6 per year and
interest is reinvested each month
5800.06
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The Number e
The exponential function with base e is called
The Natural Exponential Function
where e is an irrational constant whose value is
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The Natural Exponential Function
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The Number e
A way of seeing where the number e comes from,
consider the following example If 1 is invested
in an account for 1 year at 100 interest
compounded continuously (meaning that m gets very
large) then A converges to e
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Continuous Compound Interest
A Future value or Accumulated amount
P Present value
r Annual interest rate (in decimal form)
t Number of years
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Continuous Compound Interest
Example Find the accumulated amount of money
after 25 years if 7500 is invested at 12 per
year compounded continuously.
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Exponential Regression
Example Human population The table shows data
for the population of the world in the 20th
century. The figure shows the corresponding
scatter plot.
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Exponential Regression
The pattern of the data points suggests
exponential growth. Therefore we try to find an
exponential regression model of the form P(t)
?Abt
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Exponential Regression
We use a graphing calculator with exponential
regression capability to apply the method of
least squares and obtain the exponential model
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Nonlinear Models

• Quadratic Functions and Models
• Exponential Functions and Models
• Logarithmic Functions and Models

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A New Function
How long will it take a 800 investment to be
worth 1000 if it is continuously compounded at
7 per year?
Input
Output
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A New Function
Basically, we take the exponential function with
base b and exponent x,
and interchange the role of the variables to
define a new equation
This new equation defines a new function.
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Graphing The New Function
Example graph the function x ?2y
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Logarithms
The logarithm of x to the base b is the power to
which we need to raise b in order to get x.
Example
Answer
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Graphing y ? log2 x
Recall that y ? log2 x is equivalent to x ?2y
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Logarithms on a Calculator
Abbreviations
Common Logarithm
Base 10 Base e
Natural Logarithm
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Change of Base Formula
To compute logarithms other than common and
natural logarithms we can use
Example
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Graphs of Logarithmic Function
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Properties of Logarithms
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Application
Example How long will it take an 800
investment to be worth 1000 if it is
continuously compounded at 7 per year?
Apply ln to both sides
About 3.2 years
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Logarithmic Functions
A more general logarithmic function has the form
or, alternatively,
Example