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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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-4.5

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-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechan

ge,

degrees/hour

Numerical Techniques

Chapter 12

0 1 2 3 4 5 6-10

0

10

20

30

40

50

60Fit of Water Flow

Water Height, ft

Flow

Rate,CFS

Data

Linear FitQuadratic Fit

Cubic Fit

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

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-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

Section 12.2

Curve Fitting

There is scatter in all collected

data

!e can estimate the equation thatrepresents the data "#

$e#e"alling% a graph

There &ill "e points that do not fallon the line &e estimate

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

This line is 'ust a $"est guess%

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4/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

(east Squares

Finds the $"est fit% straight line

)inimi*es the amount each point

is a&a# from the line +ts possi"le none of the points &ill

fall on the line

(inear Regression

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

-ol#nomial Regression

(inear Regression finds a straight

line, &hich is a first order

pol#nomial +f the data doesnt represent a

straight line, a pol#nomial of

higher order ma# "e a "etter fit

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6/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

+n )T(/ #ou do "oth linear

and pol#nomial regression the

same &a# 0 the onl# difference isthe order

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7/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

pol#fit and pol#val

polyfitfinds the coefficients of a

pol#nomial representing the data

polyvaluses those coefficients tofind ne& values of #, that

correspond to the no&n values

of

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

Coefficients of the

first order

pol#nomial

descri"ing the "estfit line

2.9143* 14.2857y x= +

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

(inear Regression

3First 4rder5

6valuate ho& close a fit

#ouve achieved "# taing

the difference "et&een

the measured andcalculated points, and

adding them up

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10/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

Second 4rder Fit

7/17/2019 Ch12 Fitting Curve

11/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

fifth order pol#nomial gives a perfect fit

to 7 points

7/17/2019 Ch12 Fitting Curve

12/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturecha

nge,

degrees/hour

+mprove #our graph "# adding more

points

7/17/2019 Ch12 Fitting Curve

13/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

Section 12.89sing the +nteractive Curve Fitting

Tools

)T(/ : includes ne& interactive

plotting tools.

The# allo& #ou to annotate #our plots,

&ithout using the command &indo&.

The# include

"asic curve fitting,

more complicated curve fitting statistical tools

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14/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

9se the curve fitting tools;

Create a graph

)aing sure that the figure

&indo& is the active &indo&select

Tools

7/17/2019 Ch12 Fitting Curve

15/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

7/17/2019 Ch12 Fitting Curve

16/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

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-4

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-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

-1 0 1 2 3 4 5 6 7-20

0

20

40

60

80

100

120Some Data

Time, seconds

Tem

perature,degreesF

y = 21*x + 3.8

y = 1.1*x3- 9.3*x2+ 41*x - 3.1

data 1

linear

cubic

-lot generated using the /asic Fitting !indo&

7/17/2019 Ch12 Fitting Curve

17/23

MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturechange,

degrees/hour

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-10

-5

0

5

10

residuals

-1 0 1 2 3 4 5 6 7-20

0

20

40

60

80

100

120Some Data

Time,seconds

Temperat

ure,degreesF

y = 21*x + 3.8

y = 1.1*x3- 9.3*x2+ 41*x - 3.1

Residuals are the difference "et&een the actual and calculated data

points

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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-2.5

-2

-1.5

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-0.5

0

Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

/asic Fitting !indo&

7/17/2019 Ch12 Fitting Curve

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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-1.5

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-0.5

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Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

>ata Statistics !indo&

?ou can also access the

data statistics &indo&

from the figure menu "ar.

Select

Tools->Data Statistics

from the figure &indo&.

This &indo& allo&s #ou

to calculate statisticalfunctions interactivel#,

such as mean and

standard deviation,

"ased on the data in the

figure, and allo&s #ou to

save the results to the&orspace.

7/17/2019 Ch12 Fitting Curve

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

Curve Fitting Tool"o

n optional tool"o that allo&s

more sophisticated curve fitting

+t must "e installed on #ourcomputer

To open t#pe

cftool

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

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-1.5

-1

-0.5

0

Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

Choose the data #ou &ant to plot and anal#*e in the

Curve Fitting and >ata !indo&s

7/17/2019 Ch12 Fitting Curve

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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-2

-1.5

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-0.5

0

Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

Curve Fitting !indo&s

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MATLAB for Engineers 2E,by Holly Moore. 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write

to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

100 200 300 400 500

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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Rate of Change

time,hour

Rateoftemperaturech

ange,

degrees/hour

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980

0

50

100

150

200

250

Census Year

Po

pulation,inmillions

US Population Data

pop vs. cdate

fit 1fit 2

This plot of 9SCensus >ata &as created using the

cftool interactive modeling capa"ilit#