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HILL- Partial Derivatives REVIEW

Math Topic: partial derivatives

Key Terms: slices, tangent lines, slopes

The Math behind the Observations:

I. Background (Geometry of Partial Derivatives):

The following is taken from a website at Harvey Mudd College and provides a nice review of the geometry of partial derivatives: (See the site for more details.)

Geometrical Meaning

Suppose the graph of z = f (x, y) is the surface shown. Consider the partial derivative of f with respect to x at a point (x₀,y₀).

Holding y constant and varying x, we trace out a curve that is the intersection of the surface with the vertical plane y = y₀.

The partial derivative f_x(x₀,y₀) measures the change in z per unit increase in x along this curve. That is, f_x(x₀,y₀) is just the slope of the curve at (x₀,y₀). The geometrical interpretation of f_y(x₀,y₀) is analogous.

We now explore the above ideas in the context of our JAVA Applet for the Hill:

II. Slices:

The above screen capture shows the applet with the x-slice drawn corresponding to the point (2,1) in the xy plane. What does this picture signify?

In our screen capture we have chosen the point (2,1) in the xy plane.� The graph of f is the paraboloid

z = f (x,y) = 16 - x² -4y²

�and the vertical plane y = 1 intersects it in the parabola z = 12 - x², with y = 1 fixed. The curve in blue in the above illustrates this parabola.

III. Slopes of Tangent Lines:

In this second screen capture we note that the tangent line has been drawn in red. How does this tangent line relate to f_x?

As described in the background section, the slope of the tangent line to the parabola z = 2 - x², with y=1 fixed at the

�point ( 2,1, 8 ) is the partial derivative of f with respect to x = f_x (2,1 ).

Note: f_x(x,y ) = -2x. �When evaluated at the point we now have f_x(2,1) = -4.

Conclusion: Thus, f_x(2,1) = -4 should be the slope of the tangent line to the surface in the x-direction at the point (2,1,8).

Note: Similar observations can be made for the slice and slope of the tangent line in the y- direction. (In the applet use "Draw y-slice" and "Draw y-tangent" )

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USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil