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