Chapter 2Nonlinear Models

Sections 2.1, 2.2, and 2.3

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

The graph of a quadratic function is a parabola.

a > 0 a < 0

Quadratic Function

2( ) 0f x ax bx c a

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 graph of the revenue function is

84Maximum is at the vertex $700

2 2( 0.06)

bp

a

20.06 84R p p

The graph of the revenue function is

Maximum revenue is (700) $29,400R

20.06 84R 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

Graph of Exponential Functionswhen b > 1

( ) xf x Ab

Graph of Exponential Functionswhen 0 < b < 1

( ) xf x Ab

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

Graphing Exponential Functions

2xy

10xy

3xy

1.2xy

1xy

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

The Natural Exponential Function

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

Graphs of Logarithmic Function

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

Logarithmic Functions

( ) log bf x x C

A more general logarithmic function has the form

or, alternatively,

( ) ln f x A x C

Example: ( ) 4.6 ln 8f x x