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UTD / Math / MATH 2419 / How many variables can a function have?

# How many variables can a function have? Description

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## How many variables can a function have?

- 3.1 Functions of several variables rt = 4 costi rysunt; + 3tk 10,21] X(t) = Most ylt)=yout Elt) = 36 x lt) = -Yout ylt) = 4 cost Elt) = 3

J Crusomt) + (1 cust) + 32

## How do you write the second derivative?

= 735 dt s(t)= 5€! Don't forget about the age old question of Does dipole dipole interactions affect solubility?

StuSou

S(E)=5t

length of entre eurore

S (21) = 5(2n) = 100 If you want to learn more check out What the culture of modernity?

Curvature:

2

if yolux

a) find b) find

point where k is a maximo lim K(+)

## What is the use of partial derivatives?

koo

a) y'x) = f => Y'(x) =

f

k = 1 / 1

X

X

+

is

Find klo) =0

k'(x)= (x)"467 - x 72(x+1)"? Ex

SONY

vun

K'(x) = (x2+1 3+2) If you want to learn more check out How much money can i donate to a presidential candidate?

= 2x2 = 1

x = t J

and in

/

b) I'm kex) = 0

(BOBO BOTN)

ex) Hx,y) asin le

domain?

-12 ef sl

(x,y) } etycy

ex) flay) = x2 + y2

60 (-1 level curve

budySoup

c=0 (= 1

= 2

0 = x2 tyr 1 = x2 + y2

1 Don't forget about the age old question of What major event happened in 1848 in seneca falls ny?

ex) f(x, y) = ln (x+2y) c=0 Don't forget about the age old question of What are the methods of constitutional interpretation?
If you want to learn more check out What is limiting species’ range?

. O = ln (x+2y) 27 1= x+2y so live

13.4 Partial Derivatives.

y = f(x,y) df(x,y) = fx (x, y) = f.

and

assume

Perinative with respect to x other variables are constant 2 flxy) - lim flytax, y) = f(x,y) Tx Oxy0

Same way to find atlay

Second

derivatives with

respect

to X

os

respect to y

ex) F(x, y) - 3° +y"

, - 2x fy=2y

xx 2 fyy=2

куу я б

ex) f(x, y) = x - Xyty 2 - Sxty find where fx l fyzo

fx=2x-1-5 fy = -x +2y+ a 2x - y = 5

x 12y = -1 3y = 3 /y = 1 x=3 (31)

ex)

f(x, y) = tun (1/2)

a) Find b) check

fx/tu evaluerte (2,-2)

fxx+fyy=0

1

7 2 2

1 14

fxlx, y) = 1 to 1 fylyis) are t

r

(6) fxx = y(x2 + y 2) => 4 (-1)(x2 272) 2x = 2xy

for = x (x + y

=> X1-1) (x + y) dy =) = 2xy

+ 2xy =

or

V (1, 2) = 1000|14.06 (1-R) value of \$1000 invested for W

years at 6%

. I annual rate R-tax rate V, 1,03,26) Velos, .Zg) which has greater regative influence VII,A)= 1000 [17,06 (7) J° [145]* =7 V = 100 [16.06 []©_10(1H2)"

13.4 Total Differential Ayldx = f'(x) = dy = f(x) dx Sydr. Z= f(x,y) dz = 2 dx + 2 dy

dros

ex) Area = xy Axi f(x, y) = xy

x=2 y=3" dx = 7,0) dy = 1.02 How does this affect area.

Х.

da = 365.01

2102)

ex) z = yet

a) Find f/2, 1) f/2.1, 2.05) calculate actual difference (AZ)

f (2,1,1,05) - f (2,1) differential to Ind dz

b) use

total

03= (1.05) e? - e2 => 1.185 de ñoz

at a tot dy 7 ver, dy te dy dz= ax ay

dreal dy=-05 lilé (1) + € 6.05) => dz=1.100

velhi)

ex) use total diff. to approximate 31 Jas a) what is a function z=f(x, y) to use ?

4) what is a good starting point, IS) A pox.merde

a) = x2Jy

b) (3, 10c) * has to be close to values. az dx t az edy

27 (2x5y) dx + (x + 2 y 2) dy

- e dxcol dyas dz=((2/3) J100) I ) + (19 25 40 5) dz= 6 + 4 / 4

02= 3.1 Jas - 32 Too

dx=1 dy=-5

COSM

VILLALUI

I

so 13.5 f = f(x,y) X=XLE)

24 - 24 dx & 1 x 142

Chain Rule

Lt) afody.

at

Ex) Waxay

x=ent ya eht

a) Chain rule

dhe

Rxy let) + x (2026)

AE

kes : 21494)(82) (444) - CeBE )2 ezt

duldt

belt

hebt

y

dulde

best

b) Substitution :

I waxy =>

26

t

- et solidt- bete

ex) wz xy2 + x 2 Zty z x= { y = 2t z=2

frd duldt Goh wenys ad Chain Rules duhet w do + 2 h + 2 de o

duide- y2 + 22xz/2t) + (2xx tz? (2) + (x2+ 2zy) (6) deweld E - (46 + 27€)(2)/26 + (7/74\(65)+4)2_7 duldt 1663 + 8t + 8 => Idulde = 24 t + 8

6) Substitutieni

wat(245+ (E) (2) + (2E) (2) 2 wa + 2th rot wa 6th t&t duuld 6 = 24 t + &

012

Implicit Flx, y) 20 de

at

Differentiation

dx + 2 Qx de i

day

StudySoup

dI = Fx 746 + F duldt 03 5x dy de 16, doldt => dl dt = F/sy ex) Flx,y) = x2 2 32 - )

Fx = 2x Pyrzy dyldx = 2 2 2 2 2 dylax

Quiz)

y Ztlus (Xz) 22 22 and az

Flx, y, z) = y z

tlos (x )!

Iz . Ex Zx FZ

Zz --f Jy FZ

TH

Fxz -sin (x2) 2

FyEZ

Fzsy - X sin(x2)

23 2x

Zsin(x2)

y-sin(x Z) X

2.7 ay

-Z x-xsinlyz)

so we

Directional Derivative 3-D Slope depends on direction

have a directional derivalque u-unit direction vector (s) u= cose, tsmos 6 angle with

positive x-axis

Study Soup

Du f(x, y) = fx cose + fysine

felfy are partial

derinatives.

Quie) foxy) = xfy an x(x+y)"

And direc. derive at P(1,1) in

dissection

6 (5,4)

wnit vector

lip (1,)

Q (54)

pazu Uroll

S

| PQ = (4,3)

a=(3, &>

fx = (x+y) (1) - X(1)

Y

fx(1, 1) = 1

Aufli,1) fx cusptfysing => Ý ( 4 ) + ( 4 ) ( 2 ) = 7/20

direction

Full Dy fxy) at P(-1,2) in Q 12,6)

56 = 23,4) fx = (x2 + y2) (1) - (x-7)(3x) u 215, 15)

SOUP

fy- (x+2y3) (-1) - (xy) ay (x2= 1,2)/2 fyl-1, 2) = 7/25

Du fix,y) = 3/5 ( 2 ) + "Is (25) = 1/5

Gradient Function - If (belf) u=lose, tsinoj Duf- ofiv

of = fxir fys

NUJUTUUUJIHUUUUUUT

ex) F(x, y)= cos (1) Find Of 12,) forza sin ( 2 ) @ (2, 4) = fyz asin ( ( ) @ (2,5) = -1/2 of 7 ir 12;

Properties of zo For Flxy) maximum rate If and magnitude of

> Du f=0 of inc. is in the direction llofll. Same for minimum

except - If I-11 of 1

W

13.7 Tangent Plane since of is normal to tole (Xo Yo Zo) and of = La, b, c7= <fx,fy, fz7 7

tungent Equation of Plane: a(xx, tyly yo) + 1lz-zo) = 0

fz

Aos pris

Equation of normal lines Dorection u= Of = Laib, c7

Pont (xo, you to

X-X - yoyo z-zo

a b c XXotat ytorbt z= Zutet

CFx, Fy, F27=7F

z=f(x, y)

Flx

= f(x, y) -Z

Find equation of tan, plane and sysometic equation of

normal line. O = Z=x2+2yZ_1 @ (2,1,5) 3 x2 +1, 2+ z = 16 @ (3,-2, 3)

(1) F(x, y, z) = x2 + 2, 2-1-Z E 2 X 2 + y 2 + 2 =16=0

fx=2x fy=hy fz=1 fx = 2x fy=2y fz=2z fx=4 fy = 4 fz--| fx-6 fyzly fz=253 #f(2, 1,5) = 24,4,-1) If (3, -2,5) = 46,-4,2587 n=4ituj-k & Lab,c) niki-45+25k Point(2,1,5) = (x0 140,20) col Point (3,-2, 53) eq, = 4(x-2) + (-1)-(25)=0 eu. 6(x-3) -414-2) +253(2-53) 20

eq. of line: (x-3) (ytz) a (2-13)

b4 25

X-2

V-

7

-5

4

Aught angle

ex)

Find

cosoa lofokl

llofil

of Intersection of intersection 3x27 2 y 2 - 2 +15=0

@ (2,2,5) If = fx = lex fy = 4y fz=-1 fx=12 fy = 8 . fz=-1

wf- L12, 8,-17 @ (2,2,5)

Cost =

122 g 2 + 7

6 120g

o k= 20,0,0)

Always

13.8 our

Melatine / Absolute

e a critical

Batrenon.

point

Find d - fxx fyy- (fxy 2 at critical

i f d >o fxx >0 minimum if d70 fx x 20 maximum if dco => saddle point if d=0 test fails.

point (relagine) (relative)

ex) glx,y) = x=y-x -Y gx=2x-1 =0 x/2 gyny-1=0 y = /2

Z - - - - - - - - 2 4 - 2 = 0 col / -22, os

N

Z

gyy = -2

Gy20

d = 21-2) - Loja

d =-4 Lo

Absolute Extreme And Abs. Extrema fix, y = x 2 Uxy +5

I EXLY, Osyezl

and trikcal printi when fod

fx= 2x - 4x =0 fy = -4x = 0 x=0 v=0 z=5

& bot in range & + (x,0) = 2x x=0 no good

flo) = 4275-21

Look Vo .

along n bordes flx, o) = x2 +5 flo) = 12 +5=6

y = f(x, z) = x2 - 6x + 5

f(x, 2) = 2X-8 x=4 f (1,2) = 2 & 15--2 f (42) = 4 4 - 8 (4) +5=-11

x=1

fli,y) = 1-hyts =) Yyte flio) = 6 + (1,2)=-876=-2

X-4 f (4, y) = M-16715 => 21-16y

e f(4,0)=721 f (1,2)=-11' Absolute min is all a f(4,2) Absolute mox is al e flu, o)

00

Maximize if max

foxy) constramt glxiyl occurs at (xo iyo) : f(x, y. ) = (x, ya)

хь ца) : Д(xya )

4 47

extrema Х? Ах 2x 2

for 2)

f(x,y) = x2 + y2 contraint 2xty as - А5 3) Zx+y=s (xy) 1xtys 2y = 6) 2/л) + (h) = 5 ус уа - - -

Х=2

fly) = 2* *-S

/

50 (2, 3, 5) у 40 trker) |

Max of f(x, y, z)=xyz Luvlane of a cubel

дек, у 2) = X +y+z -4 +x - 4x 2) + - Леу 3) + - 142, 4) x+y+-4=0

Ху= А у? < x2 27 y=x x=-xy => zy x=yz -7 XFX +x - a-д х-3 уг 3 Z= 3

1x Ex, ye) : Ху? - З•337 27

yz=

from

Find un distance

(x, y) 3xt y +

Cool to 3x ty +10=0 Л. (У, -, ус уу” *, (x-o)* + (- *(x, y) : x+y" - Зүүн 10-р зел) -1, Нto vo А-2 - + - - - -

Фxe My 1x 3) Х=-3

- Лу 2 ) - -

-

Ex) Area

between

y=x

y=hx

X=2

1

26

I dy dx 27

27

2x

Tx

dx

=

X y dy dx

=)

( xy + 1/24

S2x +2 dx = x + 2 x 1 27/3]

T

ex)

Sinn 5 (1 +cosy)

- Syty cosx !

s Smxt sinx cosx dx =

s sinxltcosx) dx

e its / (Ircosx)?

uzltcosa

dyr - sinx

Sudu an

442 27

dy Sour

S S x2 + y 7 dxdy :

dx dy 7 5 4 3 arctan

dy

-tanlody =

My

=>TS

y dy

I lay l ane lu(3)

Switch

the order of integration

SudySoup

SS fex,y) dx dy

Stix,y) dx of

x-1

x

2

dx dy

Syerdy => "lzer

- 2e + /2

"

x=0 x=2

S S sing) dxdy

S schix?) dy dy

-12 cos(x)

2712(1-cosa)

Volume of solid:

x2 + 2y z tz=16

and planes

x=2

y=1

& co-ord

axis.

SS (16x2 -2,7) dy dx

S 16x - 72 dx

=

16x - 3x - 3

32.86 - 13 => 32 - 1 2 3 => 28

{{ ly + xy) dA I ly tx + 2) dydy

R= {(x,x) ) OS X = 2, 1 €4 €2}

Volumes SSE DA

of

a

pyramid..

Volume

1. Szdydz x+2y + 4 Z=12 in octant xyz70 xy plane 7=0 = 7 Xtry =12 12 . z

S 512--2yrdy dxcda 006'

first

in dA

pour hand

Z= xty x2 + y2=4 a Jux? . Z SS (x + ddidit

Volume of Zaxy z=0 x=y xl fest quadrante T x y dydx > Shqdx =>

13.3

Double

Integra's w/ Polar Coundmates

Are length=ro x = rcoso

xt year y=rsne

Id Aardride A=SS rdr de

OO

S a do 27 ago

ou lose

>

Slloso do = b ) (as de tl do

d

rardo

=)

u = cosa

du -SMO

Irrose

A

truse

can

S s sinordrdo

a

sino hr2] do

// sing

u-uh + 13

/ Soul de a fi Ar...

my 0

futuduh

1

Evaluate by converting to polar & S (x + y 2 ) by dx yo to y = Jy x2

yty24 (Circle)

Simon i do

n5 do is 4 [6] 74

55 L x xh

s xy dydx &

5 525 Z

S xydy dx

=

y + x2=75

y=0 y=x

&

5 x

q=sche

ku-coso de

.

) reest rsne

odrode

) ) cosesinor drdo

-

cosesing de

udu

78 [124] & [lasno

133 ex) p = kx y

Mass and Center of

rectunge - 10,0)

Gravity (a, o) Co, y) (a,b)

M=SSP dydx

SA

M=SS kxydydy

tudySoup

M = 5 kry 27 dx.

K623

opus

Moment about xaxis = My = J S yedAy dy " praxis = My = SS xp dy dx

?

Crater of Crawit

= (x, y)

X=

ý = m

M

toteil mass

- Ma ke

find X

Ý

P = kxy

XX

My=SSXp dydx ay

kyx dydx.

S ke 662 dx +7 Yak 62 S xdx Y2k 6 ['s x2] => ykban Mx- SSypdydy => & kxy dy dy {kx 113, p 7 dx => "3KB S x dt. Ys kb? [/text) as Ya kg az X-3kka 27/43a)

ON

14.5

Surface

Area

SA= S 2 Hy itflos do

ground y-axis

StudySoup

SA= SS S1 + f2fy? dydx a 6 f(x,y) = 12+2z-By R = {(x,y) ! x 27,2 293

319-x2 Jazz2+(-3)2 => STM 4 S Sviy dydx * If aren is a circle ALWat oo

use polar coordonates RT 3

SS STYndido Suly [lary do an S 5M z do

e

HIT

Exam) Find

S.A

z=x2 + y2

lies under a=4

Ruh

Ji + (2x +(2-3 It My 2 dy dx

do 202 21 y 2) dydzas tire rardo

2y ?

311

12_1) do

Solvalbare sa fie 2(075"I do site pour

(1), 3" CAN)

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