Surface and Held-Constant Graphs

The surface, the contour plot, the graph with $y$ held constant, $f(x, b)$, and the graph with $x$ held constant, $f(a, y)$, share one point $(a, b)$. Each graph plots the output against one input with the other input held constant. The partial derivatives at the point are the slopes of the two held-constant graphs. Shrink divides the step $h$ by 10. The secant lines on the held-constant graphs then close in on the tangent lines.

The worked column gives the linear approximation at the point. Tangent plane on the surface draws the graph of the linear approximation on the surface. A typed function is read with R’s rules. A makeFun line with two inputs works too.

Under More:

• Show. The held-constant graphs on the surface, the tangent lines, and the contour labels can each be turned off and on.
• Surface view. The turn and the tilt of the surface can be typed in degrees.
• Partial derivatives. The panel lists the formulas for the two partial derivatives of each preset.

Show R gives the R code that builds the function with makeFun() and its partial derivatives with D(). The code also evaluates the function and both partial derivatives at the point. The code then plots the contours, the surface, and both held-constant graphs.

Keys: on the contour plot, press an arrow key to move the point one step. On a held-constant graph, press the left or right arrow key to move the point along that graph. On the surface, press an arrow key to turn it. Hold Shift with an arrow key to move or turn five steps. Press + or − to zoom.