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UTD - MATH 2415 - Class Notes - Week 8

Description

Reviews

10/11/16

14.7:

Max

and

Min.

Defi

A function at (a,b) (x, y) in → Similar

z=

if some for

f(x,y) has

f(x, y) = disk with local min.

a local fla, b)

center

max for all (a, b)

r ima

of disk

disko adresine

Thm:

If the

f has

point

a local max (a, b), then

or min at If (a, b) =o.

fx=0 fy=0

><--- 4 = 6 We also discuss several other topics like Who is daniel shays?

х

(a, b) is of (a, b) =

a

3

critical

point

of

Def. A point

- f if Note: - max /min O. C. P If you want to learn more check out What is the partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture?

If you want to learn more check out What is the meaning of microglia?

☆ C.P. max/ min

10/11/16

1.) 2)

Find Set

y

of equal to o find C.P..

Recall

2nd Derivative test ( Single variable)

4= f(x), xsa is a CP. if flla) = 0 f" (a) >0 & local min. at x=a f" (a) co

local max . at x a

ex Find all local max / mins Saddle points. o f(x,y) = x + 3y’ -lexy +10x -ley +B + = 2x - (au + 10, au -ex - We also discuss several other topics like What is alkylation in chemistry?

fx = 0 = 2x -64 +10 0... 0 = -4* +4 fy = 0.64-6-6

-4 = -4* 0 + 0 :-4x = 4 5 We also discuss several other topics like What procedure would you use to separate organelles?

> x = ( 2 cp (2)

7 y = 2 JE 2 = le D(x, y) = 12-36= -24 co

> Saddle point. If you want to learn more check out What are the contributions of the chinese in san francisco?

Second Derivative Test Suppose (a,b) is a c.p. of z=f(x,y)

(ofla, b) = o)

=

-6

Let

D = det for fayl = (fas Hyy ) - (fxy)

1xx .

( Find min/max

Saddle )

ex 2: f(x,y) = x + y + 3xy +5

of = < 3x2 – 3y , 3y - 3x >

denk

(a.) Dla, b) >o (6-) Dla, b) o (c.) Dla, b) co (d.) Dla, b) = 0

htxx (a, b) o min (fxx (a, b) co max 7 Saddle point in 7 Inconclusive / No Information

2

N

CP's (0,

X7-X = 0 x(x3 - 1) = 0

> x=0, x=1 => y = 0, y = 1

.

10/13/16

Study So

D= (fxx (fyy) – (fxy).

&

fxx = 6x

tyy = ley

pt.

D(x,y) D(0,0) = -9 co > Saddle DCL, 1) = 27 >O y check fxx (1,1)

Le(1) = 6 0 7 local min.

Steps to find absolute extrema 0.) Check conditions of EVT

Lo continuous

closed, bounded set 1.) Find values of function at c.P's in D 2.) Find extrema of f on the boundary of D 3.) Compare the values in I and 2.

Lo largest abs. max. y smallest. abs. min,

o ll

expt

abs. extrema of f(x,y) = x - xy +y?

Recally

10/13/16

- Absolute Single variable Let y =

1.

Find -on

max / min extrema

thm: f(x): If f is continuous on a

closed interval La,b], then f has an abs. man and an abs. min (Extreme value theorem)

a

13

*

Def:

that

contains

ci

a closed set in IR is one all its boundary points.

A bounded set in ik is one contained within some disk

o) The f is continuous 0722 on D, which is closed & I boun ded. => abs max/min (EVT 1.) C.Di

fx= 1-4 = 0 7 4 = 1 f(2,1)

y = -x + 2y =

o x =2

IC.P: (x, y) = (2,1) 2.) On 4 . f(x,y) = f(x,0)= b

- abs Since it is inc., max on a Lib.

. emin on [0, 3) = 0

that

is

No to

NOT Closed

Extreme

Value

Theorem

for

z=f(x,y)

f

If Set may

is D &

continuous on in R, then f abs. min

a closed, bounded attains an abs.

On Li f(3,y) = 3-3y + y2 =glyd a

gly)= -3 +34 f(3, 3/2) = 3-912.701912).

=32

X=3

10/13/16

f(t) = filt) =

=

3

On

cost- sin’t. 2cos(+)(-Sin (+)) - 2 sin(t) coset) -4cosct) sin(t) = 0

> t = 1/2, 318/2, 0, it Plugin t's into f(t).

02

para

r(t) = F + tū

,o + t <3,2)

X(t) = 3+

| 4 ()= 2+ flï(t)) = 34- (3+)(2+) + (2t = 34 - 6+3 +4+?

Osts 12 = 34 -2t2 =h(t) hlt) = 3-40 =0 => t = 3/4 7 X = 174 A

y = 3/2

f1914, 312)= 4 (9)() +

Candidates 21)=1

min

f(0, 0)= 0 f(3, 2) = 1 f(3,0) = 3

+ max

exp 2: z= f(x, y) = x2 - y2

Find abs. extrema on x2 + y2 sl 0.) polynomial cont. closed, bounded set

max/min 1) vf = (2x -24 ) =0

CD: f(go)= 2.) Extrema on

0 *? +4

candidate =1

Lost <276

x= cos(t) y=sin(+)

>

Study so