Labware - MA35 Multivariable Calculus - Two Variable Calculus

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Second Partial Derivatives

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A second partial derivative is a partial derivative of a function which is itself a partial derivative of another function.

There are four types of second partial derivatives:

1. fxx(x,y) = Partial derivative of fx(x,y) with respect to x.

2. fyx(x,y) = Partial derivative of fx(x,y) with respect to y.

3. fxy(x,y) = Partial derivative of fy(x,y) with respect to x.

4. fyy(x,y) = Partial derivative of fy(x,y) with respect to y.

Third, fourth, fifth, and in general, nth partial derivatives for any positive integer n, exist as well.

Demos

Exercises

  • 1. Try entering each of these expressions for f(x,y). Note how the graphs of the second partial derivatives relate to the curvature of the graph of f(x,y):
    • 0
    • 1
    • Any constant
    • x
    • x^2/2
    • y
    • y^2/2
    • x^2-y^2 (saddle)
    • x^2 - y^4+y^2 (twin peaks)
    • xy
    • x^2/2+y^2/2+xy
    • x^2*y^2
  • 2. For polynomial functions of two variables, what are the conditions needed to yield a nonzero value for each of the following?
    • fxx(x,y)
    • fyy(x,y)
    • fxy(x,y)
    • fyx(x,y)
  • 3. a. For each of the functions in part (1), what can you say about the relationship between the mixed partials (fxy(x,y) and fyx(x,y))?

    b. Now enter the function f(x,y) = 4xy * (x^2 y^2)/(x^2 + y^2) in the second demo. Look at the curves on the first partial derivative graphs which display the second derivatives. Are any of the second partial derivatives defined at (x,y) = (0,0)? How does this differ from your observations in part (a)? (This is discussed further in the lab on equality of mixed partials.)