Directions for Functions of Two Variables:

• Wait for the applet to load and for the graph of the function to appear in the gray area above.
  (If there is no gray area, check your browser settings to make sure that Java is enabled, or try with another browser)

• The applet combines several tools for viewing functions of two variables. Use the Show menu to switch from one mode to another.

• The applet initially starts in the Input mode, which lets you choose a function to plot (you can either enter it manually, or select one from the drop-down list; click on the Plot button to create the new plot).

• In Level curves mode (select it in the Show menu) the left half of the display shows a contour plot corresponding to the 3D plot in the right half. The slider control in the lower-left corner moves a level curve highlighted in yellow on both plots.

• In Partial derivatives mode (select it in the Show menu) the right half of the display still shows the graph of the function. The lower-left corner shows a small contour plot, with a pink dot representing the point where we measure the partial derivatives. To move the point, simply click somewhere in the contour plot. There are two small graphs above the contour plot: these represent slices of the graph of the function through the given point (intersecting the 3D graph with planes parallel to the xz- and yz-planes: compare the small plots with the highlighted curves on the 3D plot). The partial derivatives are, by definition, the slopes of these graphs. Look at the bottom-right corner of the display for the values of f_x and f_y at the selected point.

• In Directional derivatives mode (select it in the Show menu) the right half of the display still shows the graph of the function. The lower-left corner shows a small contour plot, with a pink dot representing the point where we measure the directional derivative. To move the point, simply click somewhere in the contour plot. The small graph above the contour plot shows a slice of the graph of the function through the given point by a vertical plane that makes the angle theta with the -axis (theta is controlled by the slider). This same slice is shown on the 3D graph. By definition, the directional derivative is the slope of this graph. Look at the bottom-right corner of the display for the values of grad f and the directional derivative.