The partial derivatives of a function of two variables are the slopes of the slice curves.
Demos
Tangent Lines to Slice Curves
Exercises
Position the hotspot at the origin. What are the slopes of the tangent lines at this point?
Where are the slopes of the tangent lines the greatest for both slice curves?
Investigate the slopes of the tangent lines at several points on the graph. How are the slopes related to the steepness of the function graph in the x- and y-directions?
Move the hot spot to the first quadrant of the domain, then the second quandrant, third quadrant, and fourth quadrant. What can you say about the sign of the slopes of the tangent lines in each quadrant?
Investigate the tangent lines for other functions, such as f (x,y) = x2 + y2 or f (x,y) = x2 – 2xy – y2.
