1

Nonlinear ModelsChapter2

• Quadratic Functions and Models

• Exponential Functions and Models

• Logarithmic Functions and Models

• Logistic Functions and Models

Lectures 2 & 3

2

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

3

Quadratic FunctionQuadratic Function

2( ) 0f x ax bx c a

Every quadratic function has a parabola as its graph.

a > 0a < 0

4

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

5

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

6

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

7

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.

8

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

9

Graphing Exponential FunctionsGraphing Exponential Functions

Ex. ( ) 3xf x

(0,1)

( )y f x

0 1

1 3

2 9

11 3

x y

10

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

11

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

12

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

13

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

14

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

15

Continuous Compound InterestContinuous Compound Interest

rtA Pe

A = Accumulated amount P = Present valuer = Annual interest ratet = Number of years

16

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

17

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

18

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

19

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.

20

Logarithmic Function GraphsLogarithmic Function Graphs

Ex. 3( ) logf x x

(1,0)

3logy x1/ 3logy x

1/ 3( ) logf x x

(1,0)

21

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

22

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

23

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

24

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

25

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

26

Logistic FunctionLogistic Function

( ) 0, 11 x

Nf x b b

Ab

N N

b >1 0 < b <1

N is called the limiting value

27

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

28

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

29

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

.