01ns multivariable

8/19/2019 01ns Multivariable

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SRI / Calculus / Multivariable Functions 1

MULTIVARIABLE FUNCTIONS

NOTATION

Many familiar formulas

For instance:- area of a triangle

21

= A (length, l )(height, h )

- volume of a rectangular box

=V (length, l )(width, w )(height, h )

Thus we say

A : function of 2 variables

V : function of 3 variables

Notation for 2 variables is similar to that used for function of one variable),( y x f z =

Means:

z - function of x and y.

- dependent variable z is determined by specifying values for the

independent variables x and y.

NUMERICAL EXAMPLE

Quantity of beef bought

Price of beef (RM/kg)

6.00 7.00 8.00 9.00

20 2.65 2.59 2.51 2.42

40 4.14 4.05 3.94 3.88

Household income per

year (RM1000)

60 5.11 5.00 4.97 4.84

ALGEBRAIC EXAMPLE

A cylinder with closed ends has a radius r and a height h.

=V area of base x height

),(

2

hr f

hr

=

= π

×= 2 A (area of base + area of side)

),(

22 2

hr g

rhr

=

+= π π


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DOMAIN AND RANGE

A function of 2 real variables x and y, is a rule that assigns a unique real number f ( x, y)

to each point ( x, y) in some set D of the xy-plane.

The set D in this definition is the domain of the function

→ It is the set of points at which the function is defined.

If a function f is specified by a formula, so the domain of f is not stated explicitly,

Then it is understood that domain consist of all points at which the formula has no division

by zero and produces only real number;

This is called the natural domain of the function.

Common restrictions for some functions

Functions Example Restriction Remarks

rootn f , where

K,6,4,2=n0≥ f

If 0 f log (or ln) for 0 or negative

number is undefined

exponent f e no restriction f can be any real numbers


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Example 1

Let 13),(2

−= y x y x f . Find f (1, 4), f (0, 9), f ( , t ), f (ab, 9ab), the natural domain and

range of f .

2t

Solution

By substitution

1919)(3)9,(

131)(3),(

119)0(3)9,0(

514)1(3)4,1(

222

3222

2

2

−=−=

−=−=

−=−=

=−=

abbaabababab f

t t t t t t f

f

f

Because of the y , to avoid imaginary value for f ( x, y).0≥ y

},,0:),{( ℜ∈ℜ∈≥= y x y y x D

y and are positive , then2 x

}),(1:),({ ∞− y x2

x y <

Then the natural domain of f consists of all points in the region 2 x y <

},,:),{( 2 ℜ∈ℜ∈


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Example 3

Let 221),( y x y x f −−= . Find ),( 21

21 − f and the natural domain of f .

Solution

By substitution,2

212

21

21

21 )()(1),( −−−=− f

21=

Because of the square root sign, 01 22 ≥−− y x

So,

01 22 ≥−− y x

122

≤+ y x

Therefore, },,1:),{( 22 ℜ∈ℜ∈≤+= y x y x y x D

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Example 4

Let 2229),,( z y x z y x f −−−= . Find the natural domain of f .

Solution

Because of the square root sign, so f ( x, y, z) is defined only for 09 222 ≥−−− z y x

So,

09 222 ≥−−− z y x

9222 ≤++ z y x

Therefore,

},,,9:),,{( 222 ℜ∈ℜ∈ℜ∈≤++= z y x z y x z y x D


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GRAPH OF FUNCTIONS OF TWO VARIABLES

Definition 1

The set of all points ( x, y, f ( x, y)) in space, for ( x, y) in the domain of f , is called the

graph of f . The graph of f is also called the surface z = f ( x, y).

Definition 2

For a function of one variable, the graph of f ( x) in the xy- plane was defined to be

graph of the equation y = f ( x).

Definition 3

For a function of two variables, the graph of f ( x, y) in the xyz-space was defined

to be graph of the equation z = f ( x, y). In general, such a graph will be a surface in

3-space.

Characteristics of 3D Graph

Category Equation Graph

xy-plane,

( x, y, 0)

xz-plane,

( x, 0, z)Plane

yz-plane,

(0, y, z)


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Plane 1=++c z

b y

a x

b z z

a y x

==

=+

,0

,222

b yb y

a z x

=−=

=+

,

,222 Cylinder

b x

a z y

≤≤

=+

0

,222

Sphere 2222 a z y x =++


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222 y xa z −−=

Sphere

222 y xa z −−−=

22 y x z +=

22 y x z +−= Cone

22 y xa z +−=


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22 z x y +=

Cone

22 z y x +=

22 y x z +=

)( 22 y x z +−= Paraboloid

222 y xa z −−=


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22 z x y +=

Paraboloid

22 z y x +=

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Example 1

Describe the graph of the function y x y x f 211),( −−= in xyz-space.

Solution

By definition

y x z 211 −−= or 12

1 =++ z y x

By plotting the x-, y- and z- intercepts where

(1, 0, 0), (0, 2, 0), (0, 0, 1),

respectively and joining them with line segment, we will sketch a triangular portion of the

plane.

(0, 0, 1)

y

x

z

(0, 2, 0)

(1, 0, 0)


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Example 2

Sketch the graph of the following function in xyz-space.224),( y x y x f −−=

Solution

By definition, the graph of the given function is the graph of the function224 y x z −−=

By squaring both sides

4222 =++ z y x

which represents a sphere of radius 2, centered at the origin.

Since square root, , then)0( ≥ z 0422≥−− y x

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Example 3

Sketch the graph of the following function in xyz-space.222),( y x y x f +−=

Solution

The graph of the given function is the graph of the function222 y x z +−=

Since 22 y x +− , Therefore, 222),( y x y x f +−=


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Example 4

Sketch the graph of the following function in xyz-space.

x y x z 45222+−−=

SolutionThe graph of the given function is the graph of the function

y

x

z

2

54 222 =++− z y x x

By completing the square

2222

222

2222

3)2(

9)2(

52)2(

=++−

=++−

=++−−

z y x

z y x

z y x

which represents a sphere of radius 3, centered at the (2, 0, 0).

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LEVEL CURVES AND CONTOUR LINES

Definition

The set of points in the plane where a function f ( x, y) has a constant value

f ( x, y) = c is called a level curve of f .

In general, the vertical projection of the contour curve into the xy-plane is thelevel curve.

Level curves give a two-dimensional way of representing a three-dimensional

surface z = f ( x, y).

The set of level curves is known as contour map.


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Example 1

Graph and plot the level curves22100),( y x y x f −−= 0),( = y x f , and

in the domain of f in the plane.

,51),( = y x f

75),( = y x f

Solution

Domain : entire xy-plane

Range : set of real numbers 100≤

The graph is the paraboloid 22100 y x z −−=

Similarly, the level curves are the circles :

51100),( 22 =−−= y x y x f or 4922 =+ y x

75100),(22

=−−= y x y x f or 2522

=+ y x

The level curve consists of the origin alone. (It is a level curve)100),( = y x f

10

100

10

y

z

x f ( x, y) = 0

f ( x, y) = 75

f ( x, y) = 51

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Example 2

Shows some typical contour curves on the paraboloid .2225 y x z −−=

Solution

z = 0

z = 9

z = 16

z = 21

z = 25

y

z

x

y

x

z = 21

z = 16

z = 9

z = 0

x + y = 02 2

x + y = 92 2

x + y = 162 2

x + y = 252 2


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LEVEL SURFACE

The concept of a level curve for a function of two variables can be extended to functions

of three variables.

If k is a constant, then an equation of the formk z y x f =),,(

will, in general, represent a surface in 3-dimensional space (e.g.,

represents a sphere).

1222 =++ z y x

The graph of this surface is called the level surface with constant k for the function f .

y

z

x

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