Objective:
Provide a visual foundation
for partial derivatives of functions
of two variables, z = f(x,y).
Level: Calculus
II or III.
Prerequisites:
Calculus I. Derivatives of functions of a single variable together
with basic notions of surfaces of functions of two variables, z = f(x,y).
Students should be familiar with the derivatives of the functions from
first term calculus including trigonometric functions and transcendental
functions
Platform: Any
computer system with MATLAB (release 5 or later). In addition there is a
gallery of 10 animations in gif
or Quicktime format. The animations can be downloaded.
Instructor's
Notes:
As part of the motivation for the derivative
of a function y = f(x) of a single variable at a value x = a,
we often draw a set of secant lines which 'settle' into the tangent
line position. This provides a visual model for the limit process given
by
To get f '(x) we indicate that as we
vary the value of a we define a function, the derivative function,
denoted f '(x). W can think of the function f '(x) as obtaining its values
from the slopes of moving tangent line segments as shown in the following
animation.
The same arguments apply when we want to
discuss the 'partial' derivatives of z = f(x,y) with respect to
x
or y. We say, 'one of the unknowns is held fixed at a particular
value' say, x = 0.35, as in the animation at the beginning of this
demo. We then take a plane perpendicular to the x-axis at this location
and 'slice' the surface to generate a curve which has equation z = f(0.35,
y). Then the partial derivative of f with respect to y
when x = 0.35 corresponds to the slope of a tangent line moving
along this curve.
All of the verbal statements indicated
above are supposed to motivate the student to recall the appropriate visualizations
of the one variable case and then extend them to the two variable case.
Our renderings in class of the two variable case are difficult for us to
do and lack the clarity that computer graphics can bring to situations
of this type. The MATLAB routine which generated the animation at the
top of this demo, and the animations in the gallery of partials which can be
accessed by clicking here,
provide lecture tools that has been used successfully in large and small
classes for both math/science and non-science majors. The MATLAB program and the
animations in the gallery are easy to use
and let the instructor narrate the actions as they evolve on the screen.
These tools provide a visual enhancement to the usual algebraic statements that
we make when coaching students in partial differentiation techniques.
The routine partial is available
to be downloaded by clicking on partial.
A description of the routine follows together with a list of the built-in
functions available. It has the option for users to specify their own functions.
PARTIAL A graphical look at
partial derivatives of z = f(x,y).
The surface z = f(x,y) is sketched over
a rectangle in the xy-plane. The surface is then cut with a plane perpendicular
to either the x- or y-axis. The curve of intersection is displayed. Then
in a seperate figure a tangent line moves along the curve of intersection.
The x and y coordinates along with slope are displayed as the tangent line
moves along.
There are 10 built-in examples which are
accessed by using the command partial(1), partial(2), ..., partial(10),
respectively.
For general use, the form is
==> partial(f,xx,yy,typart,ptval) <==
where
f is a string with surface formula in variables x & y
xx is a vector of the form [start:incrment:end] for x-range
yy is a vector of the form [start:incrment:end] for y-range
typart is a string containing x or y to designate the type of partial
derivative
ptval is the value of x (respectively y) at which the partial is
evaluated
This routine uses MATLAB's symbolic
toolbox.
Animations for each of these functions appear in the
gallery.
Credits:
This demo, the gallery, and the MATLAB m-file were submitted by
Dr.
David R. Hill
Department of Mathematics
Temple University
and is included in Demos
with Positive Impact with his permission.
