## Functions of Two Variables

(see below for directions - read them while the applet loads!)

### Directions:

• 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). The applet starts directly with the function you will need for Problem Set 3, so you don't need to do anything here, but you are encouraged to experiment with some other functions as well (for extra practice, especially if you feel that contour plots are slightly confusing).

The 3D plot can be rotated by clicking on it and dragging the mouse (keeping the button pressed).

• Now, switch to the 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: experiment with it in order to perfect your understanding of the contour plot and how it is related to the graph of the function.

• Once you understand fairly well the contour plot, switch to the 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 fx and fy at the selected point.

• The last mode (directional derivatives and gradient vector) will only be used later in Unit 2, after these topics have been covered in lecture.