Partial Derivatives, Geometrically

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  • Partial Derivative Animation


    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)  <==
              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.

    DRH 2/14/00   Last updated 5/23/2006

    Since 3/1/2002