Functions of Several Variables

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