Optimization

I. Textbook Examples

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fty

absolutetheisAthatdeduceweAandAAvaluestheComparing

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xxx

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functionobjectiveThe

x

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50:)50(88400)(

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)4400()4400()()(4004,),(

:

250

2

10040040,

0400

xorxsoandytheneregionsthretheofwidthhet

ngrepresentissidefourtheforfencewholetheuseweIfxandythen

cagetheoflengththengrepresentisidetheforfencewholetheuseweIf

How did we know that the domain of A is [0,100]

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fty

absolutetheisVthatdeducewe

AandV

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xandxtpoinCrtical

xxxxxV

xxx

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tainconstrtheyxwhereyxyxV

functionobjectiveThe

x

364)

364(max

364)

364(264

.max)364(

0)32(0)0(,

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364()

364(2)

364(64)

364(:

3640:

)364(66)64(2)(

]32,0[;264

)264()264()(

)(642,),(

:

3

2

364

3332

2

32

2

2

326420064

xOrxthenyWhenxthenyWhen

How did we know that the domain of A is [0,32]

wayttheallswimwhenistimetravelthesThatabsolutetheishourTthatdeducewe

TandT

TvaluestheComparing

orsThat

andthatsuchtpoinCrtical

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functionobjectiveThe

min,'min1)(

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sin)(05.13

0)0(,

23.1368.1

296sin)(:

68.1)180(9696'

],0[32

2cos:

31

2cos21)(

]],0[;32

sin

],0[;32

2sin2

)(

:

0

0

00

0

0:

),(

,,,

:0

Thusngmiswidoneistripwholethethe

tpoinfinishthewithcoincidespreacheswhen

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walkingdoneistripwholetheandngmiswinoistherethatmeansWhich

thentpoinstartthewithcoincidesptpointheIf

How did we know that the domain of A is [0,π]

ftaboutissconditiuongivenhesatisfyingladdershortesttheThus

L

absoluteanbemustitthenonextremumlocalonlytheisfSince

localatheisxthatdeducewe

onxfandonxf

xorxnotwhyxtpoinCrticalxxxxxx

xx

xxxxxf

xxxxbngsubstituti

xxx

xxbwherebxxf

functionobjectiveThe

x

992.14

992.14)916

641()334(

min),,0()34(

.min34

),34(0)()34,0(0)(

)30(34192:

)192)(3(2]64)64(364[)3(2

)3(2]641[)64)(2()3()(

]641[)3(

)3(8,))3(8()3(

83

;)3()(

:

323

34

3

3

33

33

3

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3

232

22

22

22

3

II. Other Examples

1. Homework Examples: ( Section 4.4 Exercises )a. Problems: 9 (Maximum product) and 11 (Minimum sum)b. Problems: 19 ( Minimum distance)

2. Previous Exams Problems:a. Q4 of Exam 2 ,Fall, 2013.b. Q8 of Exam 2, Fall, 2012

3. Questions done in the homework and quizzes in MyMathLab ‘