Functions of Several Variables

Algebraic

A real-value function of f of x, y, z, … is a rule for manufacturing a new number, written f(x, y, z …), from the values of a sequence of independent variables (x, y, z, …) . f ( x, y ) x 2 y 2 4 x 1

Example: Function of 2 variables

For the function f, evaluate.

f (2,1) 22 12 4 2 1 2

Substitute 2 for x and 1 for y.

2

2

f (a, 5) a 5 4 a 1 a 2 4a 26

f ( x h, y k ) x h y k 4 x h 1

2

2

Example: Function of 3 variables

For the function f, evaluate.

Substitute a for x and 5 for y.

Substitute x+h for x and y+k for y.

g ( x, y, z ) ln( x 2 y z )

g (1, 0, 0) ln 1 2 0 0 ln 1 0 g ( y, z, x) ln y 2 z x ln y 2 z x

Functions of Several Variables

Numeric

Use the tabular representation of the function f to evaluate the function.

f ( x, y ) x 0.8 y 1.5 xy f (30,10) 30 0.8 10 1.5 30 10 472 x y

y

10

10

20

20

x

10

10

152

294

20

20

312

604

f (10, 20) 10 0.8 20 1.5 10 20 294

30

30

472

914

Functions of Several Variables

The graph of the function f of two variables consists of all points of the form

(x, y, f(x, y)) .

Graphical

In 3d space, there are 3 mutually perpendicular axes:

• x-axis (extend to the front)

• y-axis (extend to the right)

• z-axis (extend upwards)

In 3d space, each point has three coordinates:

• x-coordinate

• y-coordinate

• z-coordinate

Functions of Several Variables

In 3d space, there are 3 coordinate planes:

• xy-plane where z = 0

• xz-plane where y = 0

• yz-axis where x = 0

Graphical

Functions of Several Variables

Graphical

If we hold a variable constant, the graph of the set of points is a plane.

Sketch y = 2

Since y is a constant, 2, we can think of set of points as all points

• 2 units right of the xz-plane and

• extends indefinitely in the x direction

• extends indefinitely in the z direction

Functions of Several Variables

Sketch z = - 2

All points 2 units below the xy-plane…