Multivariable Functions, Level Curves and Partial Derivatives
Domain and Range for Multivariable Functions
The function z = f (x , y) is a function of two variables with
dependent variable ‘z’ and independent variables ‘x’ and ‘y.’
The domain of z = f (x , y) is the two‐dimensional set of all points in the xy plane
which are valid inputs into the function.
The range of z = f (x , y) is the set of all ‘z’ outputs plotted to make the surface.
Ex) For the function f (x , y) = y 2 − x , sketch the domain set.
Based on this domain, what range of outputs would you get for ‘z’ ?
Ex) Sketch the domain of the function f (x , y) = ln(16 − x 2 − y 2 ) .
What would its range of ‘z’ outputs look like?
Level Curves
One way to visualize what a function’s graph might look like is to create a
topographic projection of the function inside its domain region.
These topographic projections are called level curves or contour curves for the
function z = f (x , y) .
Ex) Sketch the level curves for the function f (x , y) = y 2 − x using z = 0, 1, 2 and 3.
Using these level curves, try to piece together a visualization of the surface graph.
Ex) Sketch the level curves for the function f (x , y) = e −( x +y ) .
Choose appropriate ‘z’ values to use to make your level curves.
2
2
Using these level curves, try to piece together a visualization of the surface graph.
For the test you may be asked to sketch the domain of a function of two variables
but you will need to match its level curves to its surface graph.
Partial Derivatives
To analyze the ‘slopes’ at various points on a 3D function’s surface we will use
partial derivatives. A function can have as many first partial derivatives as it has
independent variables.
The partial derivative with respect to x
measures the rate of change (or slope) in
the increasing ‘x’ direction along a surface.
This treats ‘y’ as a constant.
∂z
, z x or fx .
Denoted as either
∂x
The partial derivative with respect to y
measures the rate of change in the increasing
‘y’ direction along a surface.
This treats ‘x’ as a constant.
∂z
, zy or fy .
Denoted as either
∂y
Ex) Find the first partial derivatives for the following functions:
a) f (x , y) = 4 x 3 − 6 xy 2 + 2y − 7
b) z = e x sin(xy)
w 2v 2
c) R(w ,v) = 2 2
w +v
Evaluating Partial Derivatives
When evaluating a partial derivative these notations may be used:
∂z
∂z
f x ( x0 , y0 )
( x0 , y0 )
∂ x x = x ,y = y
∂x
0
0
∂R
w 2v 2
(2, −1) for the function R(w ,v) = 2 2 .
Ex) Evaluate
w +v
∂v
More than 2 Independent Variables
You’re not restricted to functions only using 2 independent variables …
Ex) Find the first partial derivatives for the 3‐variable function
w = a2 + b2 + c2
.
Second Partial Derivatives
There are only 2 first partial derivatives for a function of 2 variables. There are a
total of 4 second partial derivatives.
The ‘Regular’ Second Partials
∂2 f
f xx = 2
∂x
∂2 f
f yy = 2
∂y
The ‘Mixed’ Second Partials
∂2 f
f xy =
∂x ∂y
∂2 f
f yx =
∂y ∂x
The subscript notation is helpful here because it tells you the order in which to get
to that second partial.
Ex) Find all 4 second partial derivatives for the function z = x 4 cos(5y) + 3xy 2 .