chapter 2 nonlinear models sections 2.1, 2.2, and 2.3
TRANSCRIPT
Nonlinear Models
Quadratic Functions and Models
Exponential Functions and Models
Logarithmic Functions and Models
Quadratic 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
Example:
where a, b, and c are fixed numbers
2( ) 12 3 1f x x x
Vertex coordinates are:
x – intercepts are solutions of
y – intercept is:
symmetry
,2 2
b bx y f
a a
2 0ax bx c 2
bx
a
0x y c
Vertex, Intercepts, Symmetry
Vertex:
x – intercepts
y – intercept
21 ( 1) 9
2 2
bx y f
a
0 8x y
2( ) 2 8f x x x
2 2 8 0x x 4,2x
Graph of a Quadratic FunctionExample 1: Sketch the graph of
Vertex:
x – intercepts
y – intercept
12 3 (3 / 2) 0
2 2 4 2
bx y f
a
0 9x y
24 12 9 0x x 3/ 2x
Graph of a Quadratic FunctionExample 2: Sketch the graph of 2( ) 4 12 9f x x x
Vertex:
x – intercepts
y – intercept
4 (4) 42
bx y f
a
0 12x y
214 12 0
2x x
Graph of a Quadratic FunctionExample 3: Sketch the graph of 21
( ) 4 122
g x x x
no solutions
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.
( ) 3 600q p p
( ) 3 600R p pq p p 23 600p p
Maximum is at the vertex, p = $100
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?
Applications
The annual revenue is given by
Solution
( ) 0.06 84R p pq p p 20.06 84p p
The annual cost as function of q is given by
( ) 20000 20C q q
The annual cost as function of p is given by
( ) 20000 20 20000 20 0.06 84
1.2 21680
C p q p
p
Thus the annual profit function is given by
Solution
2
2
( ) ( 0.06 84 ) 1.2 21680
0.06 85.2 21680
P p R C p p p
p p
The profit function is 2( ) 0.06 85.2 21680P p p p
85.2Maximum is at the vertex $710
2 2( 0.06)
bp
a
The profit function is 2( ) 0.06 85.2 21680P p p p
Maximum profit is (710) $8,566
Corresponding Revenue is (710) $29,394
P
R
Nonlinear Models
Quadratic Functions and Models
Exponential Functions and Models
Logarithmic Functions and Models
( ) 0 and 1xf x b b b
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,
>0 for all xy b x
Exponential Functions
Exponential Functions
Example: ( ) 5 3xf x
where A is an arbitrary but constant real number.
We will consider the more general exponential function defined by
( ) 0 and 1xf x Ab b b
2xy
x y-4 1/16
-3 1/8
-2 1/4
-1 1/2
0 1
1 2
2 4
3 8
x y-4 1/16
-3 1/8
-2 1/4
-1 1/2
0 1
1 2
2 4
3 8
Graph of Exponential Functionswhen b > 1
1
2
x
y
Graphing Exponential Functions
x y-3 8
-2 4
-1 2
0 1
1 1/2
2 1/4
3 1/8
4 1/16
x y-3 8
-2 4
-1 2
0 1
1 1/2
2 1/4
3 1/8
4 1/16
Laws 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
Finding the Exponential Curve Through Two Points
Example: Find an exponential curve y Abx that passes through (1,10) and (3,40).
110 Ab340 Ab
340
10
Ab
Ab
24 b2b
Plugging in b 2 we get A 5
( ) 5 2 xf x
2b
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.
0.4479( ) 20 4 tQ t
Thus, after 6 hours there are about 830 bacteria
Exponential Functions-Examples
0.4479(6)(6) 20 4 829.86Q When t 6
Compound Interest
( ) 1mt
rA t P
m
A = the future value
P = Present valuer = Annual interest rate (in decimal form)m = Number of times/year interest is compoundedt = Number of years
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
Compound Interest
where e is an irrational constant whose value is
2.718281828459045...e
The exponential function with base e is called “The Natural Exponential Function”
( ) xy f x e
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:
11
m
A em
The Number e
Continuous Compound Interest
rtA Pe
A = Future value or Accumulated amount P = Present valuer = Annual interest rate (in decimal form)t = Number of years
Example: 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 Interest
Example: Human population The table shows data for the population of the world in the 20th century. The figure shows the corresponding scatter plot.
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
Exponential Regression
We use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model (0.008079266) (1.013731)tp
Exponential Regression
Nonlinear Models
Quadratic Functions and Models
Exponential Functions and Models
Logarithmic Functions and Models
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
A good guess for is 3.187765t t
0.075
4te
A New Function
Input Output
Basically, we take the exponential function with base b and exponent x,
xy band interchange the role of the variables to define a new equation
This new equation defines a new function.
A New Function
0 y xx b
2yx x y
1/16 1/16
1/8 1/8
1/4 1/4
1/2 1/2
1 1
2 2
4 4
8 8
x y1/16 -4
1/8 -3
1/4 -2
1/2 -1
1 0
2 1
4 2
8 3
Graphing The New Function
Example: graph the function x 2y
Logarithms
log if and only if 0yby x x b x
The logarithm of x to the base b is the power to which we need to raise b in order to get x.
Example:
3
7
1/ 3
5
log 81
log 1
log 9
log 5
3
7
1/ 3
5
log 81 4
log 1 0
log 9 2
log 5 1
Answer:
2yx x y
1/16 1/16
1/8 1/8
1/4 1/4
1/2 1/2
1 1
2 2
4 4
8 8
x y1/16 -4
1/8 -3
1/4 -2
1/2 -1
1 0
2 1
4 2
8 3
Graphing y log2 x
Recall that y log2 x is equivalent to x 2y
Common Logarithm10log log
ln loge
x x
x x
Natural Logarithm
Abbreviations
log 4 0.60206
ln 26 3.2581
Base 10
Base e
Logarithms on a Calculator
To compute logarithms other than common and natural logarithms we can use:
log lnlog
log lnba a
ab b
9log15
log 15 1.232487log9
Example:
Change of Base Formula
Properties of Logarithms
log log log
log log log
log log
log 1 0
l
1.
2.
3.
4
. g5 o 1
.
b b b
b b b
nb b
b
b
mn m n
mm n
n
m n m
b
Example: How long will it take an $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