1.Multivariable Functions

8/12/2019 1.Multivariable Functions

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Syahirbanun Isa , FSTPi, UTHM

Function of single variable( ) y f x 3 5 y x ( ) ln x f x x e

( )u g v 1u v

Function of multiple variablesTwo variables Three variables

( , ) z f x y

23 z x y

3( , ) 2 1 f x y xy x

( , )V f r h

2V rh

( , , )w f x y z

2 2 2w x y z

( , , ) ln( ) f x y z xyz

( , , )u g v w t

414u v t w


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Domain and rangeFunction of single variable (recall)

Domain Consisted all values of x that is possible

Range The set of y values when x varies over the domain

ExampleGiven 2 y x

2 y x Domain :Range :

{ : } D x x R

{ : 0, } R y y y R


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Common restriction for function

Type offunction Example Restriction Remarks

Root

Reciprocal

log or ln

,

2,4,

n a

n

1

a

log( )

ln( )

a

a

0a

0a

0a

The function willbecome complexnumber if .

Any number dividedby zero is undefined.

0a

log or ln for

is undefined.

0a


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Function of multiple variables

Domain Consisted all values of x , y and z that is possible

Range The set of f values when x , y and z varies overthe domain

Let function of two variables is f ( x, y) and three variables is f ( x, y, z )

Example: Domain and range
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Level curves and contour linesLevel curves Two dimensional curves with

equation where c is anyappropriate number.

( , ) f x y c

Contour lines Set of level curves

level curves


7/20Syahirbanun Isa , FSTPi, UTHM

Equation NoteCircle

r 2 2 2

( ) ( ) x h y k r Centre: (h,k )Radius: r(h,k )

ab

(h,k )

2 2

2 2

( ) ( )1

x h y k a b

Centre: (h,k )Distance x: aDistance y: b

Ellipse

Parabola(i) (ii)

(i)

(ii)

2( ) y a x h k 2( ) x a y k h

Vertex: (h,k )

2 2

2 2

( ) ( )

1

x h y k

a b

2 2

2 2

( ) ( )1

y k x ha b

(i)

(ii)

(h,k )

Centre: (h,k )Hyperbola

Example: Level curves and contour lines


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Limits and continuity

Limits( , , ) ( , , )lim ( , , ) x y z a b c f x y z L

Two variables

Three variables

( , ) ( , )lim ( , )

x y a b f x y L

Finding the limits

Determine f (a ,b) or f (a ,b,c) define or not.If defined,

or( , ) ( , )lim ( , ) x y a b f x y L ( , , ) ( , , )lim ( , , ) x y z a b c f x y z LIf not

2nd

attemptSimplify f . After simplify, check the new

f defined or not at the point.

If cant simplify or

f notdefined



or

The limit does not exist. Show that the limit doesnot exist by choosing any smooth curves whichpass through the point.

Example: Limits

Continuity f is defined at point (a ,b) or (a ,b,c)

Limits of f is exist

( , ) ( , )lim ( , ) ( , )

x y a b f x y f a b

( , , ) ( , , )lim ( , , ) ( , , )

x y z a b c f x y z f a b c

1

2

3

Example: Continuity
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Implicit partial differentiation

Implicit function?? 9 cos( ) xy xy x 2 1 0 x ye

10 10 tan( ) 3sin zx xyz x If given an implicit function and let .( , , ) 0 F x y z Thus,

x x

z

F z F and

y y

z

F z F

Example: Implicit differentiation


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Chain ruleTwo variables Three variables

If z = f ( x, y) is a functionof x and y

And ( ) x g t ( ) y h t

Thendz z dx z dydt x dt y dt

If w = f ( x, y,z ) is a functionof x,y and z

And ( ) x g t ( ) y h t

Thendw w dx w dy w dz dt x dt y dt z dt

( ) z l t

Example: Chain rule


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Application of partial derivatives

Tangent plane

Slope of surface

Rate of change Small increment andapproximation

Analysis of error

Local extreme value


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Tangent plane

Tangent plane to the surface z = f ( x, y) at point is0 0 0( , , ) x y z

0 0 0 0 0 0 0( , )( ) ( , )( ) x y z z f x y x x f x y y y

Example: Tangent plane

Slope of surface

Slope of surface in x direction at 0 0( , ) x y

0 0( , ) x f x y

Slope of surface in y direction at 0 0( , ) x y

0 0( , ) y f x y0 0( , ) x y

Example: Slope of surface

0 0 0 0 0 0 0 0 0 0 0 0( , , )( ) ( , , )( ) ( , , )( ) 0 x y z f x y z x x f x y z y y f x y z z z
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Rate of change

If change of z = f ( x, y) in time, z dx z dy x dt y dt

the rate of change

dz dt

Example: Rate of change

Small increment and approximation

Total differential

Two variables

Three variables

x ydz f dx f dy

x y z dw f dx f dy f dz


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Example: Small increment and approximation

Please consider these 4 things

Approximate change, dz a

Exact change, dz

Approximate value, z a

Exact value, z

Use total differential to find dz a

1 1 0 0( , ) ( , )dz f x y f x y

0 0( , )a a z f x y dz

1 1( , ) z f x y

Approximation of using total differential

Exact calculation

If given two points, and ;0 0( , ) x y 1 1( , ) x y


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Analysis of error

Let z = f ( x, y)

Error of z

Relative error of z

Percentage of error

| | | | | | x ydz f dx f dy

x ydz dx dy

f f z z z

Example: Analysis of error


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SyahirbanunIsa , FSTPi, UTHM

Local extreme valueHow to determine local extreme value(s)???

Step 1 Find and, , , x y xx xy f f f f yy f

Step 2 Let and , then find all critical point(s)0 x f 0 y f

Step 3 2( , ) ( ) xx yy xyG x y f f f FindStep 4 Calculate G for each critical point. Do a conclusion.

ConclusionLet (a ,b) is critical point

( , ) 0G a b ( , ) 0G a b ( , ) 0G a b

( , ) 0 xx f a b ( , ) 0 xx f a b z has localmax at (a ,b)

z has localmin at (a ,b)

z has saddlepoint at (a ,b)

Noconclusion

Example: Local extreme value

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1.Multivariable Functions