Differentiation of Real-Valued Functions

startsection section10mm-.5 Discussion 14 Contour Plots

 

In this discussion, we will investigate a single contour plot in order to determine what we can learn about a function from its contour plot. First, we must define what we mean by a contour plot.

The contour curve or level set of a function $f: {\cal D} \subset {\bf R}^{2} \rightarrow {\bf R}$ for the value c, is the subset of ${\cal D}$, given by

$$\{ (x,y):(x,y) \in {\cal D} \mbox{ and } f(x,y) = c \}.$$

A contour plot for f is a plot of a portion of the domain of f showing several level sets of f for different values c.

Usually the collection of values c that correspond to the level sets in a contour plot are evenly spaced values, for example, 5, 10, 15, 20, and so on. The values c for the level curves are indicated on the plot.

startsection subsection10mm.5 Exercises These exercises refer to the accompanying contour plot of Doublehead Mountain in New Hampshire. In this case, the function f under consideration is altitude above sea level measured in feet. Of course, there is no symbolic form for such a function. Note that the contours on the plot are for values c that are separated by 20 feet.

1.

Locate the local maximum values for the altitude function on the plot. How would you characterize the contours near the local maxima? Will your characterization always hold near a maximum? Explain why or why not. If not, explain under what circumstances your characterization applies.

2.

In addition to Trails 202 and 204 to the summit of North Doublehead Mountain, we can imagine a straight-line path from the junction of Trails 202 and 204 just below the 1900 foot contour heading directly to the summit of North Doublehead Mountain. Comparing this new path to Trail 202, which would you prefer to take and why? What features of the contour plot indicate that this is a preferred route?

3.

Let Q be the point between North Doublehead and South Doublehead where Trail 204 ends on Trail 202.

(a)

Carefully describe the path along Trail 202 from North Doublehead through the point Q to south Doublehead.

(b)

Suppose Trail 204 continued through the point Q to the point P marked on the eastern edge of the plot. Describe the path along this new trail from the point just below the 1900 foot contour where Trails 202 and 204 meet, through the point Q to the point P.

(c)

What does the surface of Doublehead Mountain look like around the point Q?

  

Figure: A contour plot of the region near Doublehead Mt. in New Hampshire. Notice that the contours correspond to a 20 foot change in altitude and the 100 foot contours are labeled on the plot.

startsection section10mm-.5 Discussion 15 Plotting Functions of Two Variables in Maple

 

This is a L^ATEXversion of a Maple worksheet. The worksheet is available in the directory /home/stu/courses/math141.

startsection subsection10mm.5 Introduction

This discussion will introduce the Maple commands for plotting functions of two variables that we will be using in this course. These commands will allow us to plot complicated functions with ease. However, we want to emphasize that it is not sufficient just to be able to plot functions, we must also be able to interpret the plots.

The first part of this discussion introduces the commands and the second part of the discussion asks you to use these commands to analyze several functions of two variables.

startsection subsection10mm.5 Plotting Commands for Functions of Two Variables

As we have seen previously, the plotting commands that we will be using are contained in the Maple package ``plots''. In order to use the commands in this package, you must load it at the beginning of your Maple session using the ``with'' command. This command is given below. Enter this command before continuing.

    with(plots):

We will use plot commands, contourplot, plot3d, and contourplot3d . (Words in green are hyperlinks to the corresponding help pages.) The first two display 2-dimensional plots of the domain of the function. The second two display 3-dimensional plots of the graph of the function. The syntax of each of the plot commands is similar. To use each of these commands we will first define the function we wish to plot in Maple. We do this using the := construction, for example, the function f(x,y) = x²-y²is defined using the command:

f:=(x,y)-> x^2-y^2;

After defining the function, we can use the plot commands below to generate the different graphical representations of f.