An introduction to plotting with Maple Worksheet by Mike May, S.J.- maymk@slu.edu Edited by Russell Blyth - blythrd@slu.edu QyRJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHIiIi In previous courses you have used a graphing calculator to graph functions of one variable. Having a convenient grapher makes the concepts of calculus easier to understand. However, most of Calculus III deals with functions of several variables and our calculators are not very good at graphing surfaces. The point of this worksheet is to give an introduction to graphing with Maple. This should make it easy for you to produce the relevant pictures for this course.

Since we are more familiar with functions of one variable, we start with them. The syntax for plotting a function of one variable is plot("function of x", x="low value of x".."high value of x"); where the expressions in quotation marks are to be replaced by appropriate values. Hit enter to execute the command below. QyQtSSVwbG90RzYiNiQtSSRzaW5HRiU2I0kieEdGJS9GKjssJEkjUGlHJSpwcm90ZWN0ZWRHISIiRi4iIiI= Things to notice: The "function of x" is not an equation. It is what we would set y equal to if we were writing an equation. We specified the x-range we wanted, and Maple decided on an appropriate y-range. We will learn how to specify the y-range a little later. Click once on the graph. The graph should now be enclosed in a box. Dragging the corners of the box lets you resize the graph. The second row of icons above has also been replaced by a new row. The new row has buttons that control the style of the graph. From left to right you should see a box with the coordinates of the point the cursor is located over (or the coordinates of the last point the cursor was over before your cursor left the plot box), a drop down menu of icons that control the style of the graph, a drop down menu of icons that control the style of the coordinate axes, a "1:1" button that determines whether the x and y axes need to have the same scale, a drop down menu of icons that controls the behavior of the mouse (point probe, scale (zoom the graph by dragging), or pan (translate the graph by dragging)) , and a button for whether or not to show the grid for the graph.

1) Plot the graph of y = cos(2*x) with NiMxLCQlI1BpRyEiIiUieEc= and NiMxJSJ4RyomIiIjIiIiJSNQaUdGJw== Stretch the graph to cover the screen. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn 2) Replot the graph above, but modify the style so that it is plotted with dots rather than with connected lines, and so that the axes are on the sides of the graph rather than in the middle of them. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn 3) Replot the graph above, but modify the style so that it is a connected curve, with no axes, and the same scale for the x and y axes. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

We would like to be able to specify the y-range of a plot. If a plot has a vertical asymptote, Maple will give such a large y-range that we miss interesting details. The solution is to specify the y-range. Specifying the y-range is done using the same conventions used to specify the x-range. The command plot("function of x", x="low value of x".."high value of x"); becomes plot("function of x", x="low value of x".."high value of x", y="low value of y".."high value of y"); For an example, execute the plotting commands given below. QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJi1JJGNvc0dGJTYjSSJ4R0YoIiIiRi4hIiIvRi47ISM1IiM3Ri8= QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUqJi1JJGNvc0dGJTYjSSJ4R0YoIiIiRi4hIiIvRi47ISM1IiM3L0kieUdGKDshIiMiIiNGLw== The second graph is much more informative than the first about the behavior of this function. It is worthwhile at this point to note where we have been using imprecise language. By "x-range" we mean "the range of the independent variable", while "y-range" is "the range of the dependent variable". Equivalently, x-range and y-range respectively stand for the ranges of the input and output variables. The examples below show what happens when we change the names of the variables. Note in particular the variable labels on the axes. QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUqJi1JJGNvc0dGJTYjSSJ4R0YoIiIiRi4hIiIvRi47ISM1IiM3L0kiekdGKDshIiMiIiNGLw== QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUqJi1JJGNvc0dGJTYjSSJ5R0YoIiIiRi4hIiIvRi47ISM1IiM3L0kieEdGKDshIiMiIiNGLw== QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUqJi1JJGNvc0dGJTYjSSJ4R0YoIiIiRi4hIiIvSSJ5R0YoOyEjNSIjNy9GLjshIiMiIiNGLw== We can use any letters for the input and output variables, but the input range is specified before the output range or we get an error.

It is also useful to be able to see the graphs of several functions on the same graph. In Calculus I and II we were interested in seeing the graph of a function and its derivative at the same time. We also wanted to compare a function with the polynomial approximations we obtained for it. To plot several functions at the same time, use the command plot({"function1 of x", "function2 of x", "etc."}, x="low value of x".."high value of x"); Note that the set of functions is enclosed in braces. For an example, consider the following commands which plot cos(x) along with some of the Taylor polynomial approximations of cos(x). QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JiIiIiwoRitGKyomI0YrIiIjRispSSJ4R0YoRi9GKyEiIiomI0YrIiNDRispRjEiIiVGK0YrLUkkY29zR0YlNiNGMSwmRitGK0YtRjIvRjE7ISIkIiImRis= QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU8JiIiIiwoRitGKyomI0YrIiIjRispSSJ4R0YoRi9GKyEiIiomI0YrIiNDRispRjEiIiVGK0YrLUkkY29zR0YlNiNGMSwmRitGK0YtRjIvRjE7ISIkIiImL0kieUdGKDskISM6RjIkIiM6RjJGKw== %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

4) Produce a nice plot of the graph of y = tan(x) with x between 0 and 10. You will need to specify the y-range. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn 5) Plot 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 with NiMxLCQiIiMhIiIlInhH and NiMxJSJ4RyIiIw==, along with the lines tangent to that graph when x = -1.5, -.5, .5, and 1.5. Crop the graph to show the region where y is between -2 and 4. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

As was noted earlier, you can graph a function of one variable on a graphing calculator. Often, a calculator is the easier way to deal with such graphs. There are two reasons to look at them with Maple when your calculator is already at hand: It gives an introduction to graphing with Maple using familiar material. (This is the reason we include it here.) Maple graphs can easily be printed if you want to write on them or turn them in.

The syntax for plotting the graph of a function in two variables is similar to the one variable case. We use the command: plot3d("function of x and y", x="low x".."high x", y="lowy".."high y"); as we see with the following command. QyQtSSdwbG90M2RHNiI2JSwmLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInhHRiUiIiItRik2I0kieUdGJUYvL0YuOyEiJCIiJi9GMjssJEkjUGlHRishIiIsJComIiIjRi9GOkYvRi9GLw== Once again, if you click once on the graph, it will be enclosed in a box, and a set of buttons will appear up on the menu bar. There are 2 boxes labeled NiMlJnRoZXRhRw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5ODE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== with scroll arrows for specifying the viewing angle, a drop down list of 7 buttons for style used on the graph itself, a drop down list of 4 buttons for style used on the axes, a "1:1" button to use the same scale on all 3 axes, and a drop down list of 3 possible behaviors (rotate, scale, and pan) of dragging the mouse on the plot. The default plot uses a wire frame model, that is opaque (the front bump hides the back bump), the axes are not displayed, from a viewing angle of 45 degrees on both angles (the next section describes how to interpret and control the viewing angles).

6) Redraw the graph given above. Then use the buttons to produce an image that: a) shades the image, draws a grid of lines on the image, and boxes the axes around the image; 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 b) shades the image and draws contour lines on it, while putting the axes in standard position through the origin; 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 c) uses a wire frame model with contour lines, and suppresses the axes. QyQtSSdwbG90M2RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JSwmLUkkc2luR0YlNiNJInhHRigiIiItRiw2I0kieUdGKEYvL0YuOyEiJCIiJi9GMjssJEkjUGlHRiYhIiIsJComIiIjRi9GOkYvRi9GLw==

The list of controls for changing the appearance of a graph includes controls for theta and phi, the angle coordinates (in degrees) of the position in spherical coordinates that the graph is viewed from. The angle phi is the angle to the positive z-axis. Thus an angle of NiMvJSRwaGlHIiIh looks down on the x-y plane from above, NiMvJSRwaGlHIiMhKg== gives a viewpoint looking along the x-y plane, while NiMvJSRwaGlHIiQhPQ== looks up from below. The angle theta measures the rotation in the x-y plane. If NiMvJSRwaGlHIiIh (so we are looking down the z-axis), then NiMvJSZ0aGV0YUciIiE= shows the y-axis horizontal increasing from left to right and the x-axis vertical decreasing as you go up. The default setting has NiMlJnRoZXRhRw== and NiMlJHBoaUc= both set to 45. 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

7) Redraw the graph above. Adjust the viewing angle so that theta = 60 and phi = 30. Use a wireframe model that is opaque, and have the axes on the outside of the graph. 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 8) Find the angle settings that look down the z axis and put the x-y plane in standard position for the plane. (Standard position has the x-axis horizontal and increasing to the right. The y-axis is vertical and increasing as you rise.) 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 The correct settings are..... 9) View the graph above as a contour map with the x-y plane in standard position. 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Thus to get the map above with contours corresponding to the values of -2/3, 0, 1/4, and 5/4 we use the command: plots[contourplot](sin(x) + sin(y), x=-3..5, y= -Pi..2*Pi, contours= [-2/3, 0, 1/4, 5/4]); 10) Redraw the contour plot you found in the previous exercise, using the plot options rather than adjusting the controls and redrawing. 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The syntax for showing more than one graph on the same plot is similar to the one variable case. We simply replace the function with a set of functions. 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 The control of the z-range is done with a view = "low value of z".."high value of z" clause inserted into the plot3d command. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiVGKy1GIzYlRistRiM2JS1GLDYlUSdwbG90M2RGJ0YvRjItSSNtb0dGJDYwUTAmQXBwbHlGdW5jdGlvbjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGWC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JKG1mZW5jZWRHRiQ2JC1GIzYrRistRlxvNiYtRiM2J0YrLUYjNidGKy1GIzYlLUYsNiVRJHNpbkYnL0YwRkZGQkY+LUZcbzYkLUYjNiMtRiw2JVEieEYnRi9GMkZCLUY/NjBRIitGJ0ZCRkRGR0ZJRktGTUZPRlFGUy9GV1EwbWVkaXVtbWF0aHNwYWNlRicvRlpGZ3BGZW5GaG4tRiM2JUZob0Y+LUZcbzYkLUYjNiMtRiw2JVEieUYnRi9GMkZCRistRj82MFEiLEYnRkJGRC9GSEYxRklGS0ZNRk9GUUZTRlYvRlpRM3Zlcnl0aGlja21hdGhzcGFjZUYnRmVuRmhuLUYjNihGKy1GIzYlRistSSVtc3VwR0YkNiVGYHAtSSNtbkdGJDYkUSIyRidGQi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0ZjcC1GIzYlRistRl1yNiVGX3FGX3JGY3JGKy1GPzYwUSgmbWludXM7RidGQkZERkdGSUZLRk1GT0ZRRlNGZnBGaHBGZW5GaG4tRmByNiRRIjVGJ0ZCRitGQi8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGYnEtRiM2JkZgcC1GPzYwUSI9RidGQkZERkdGSUZLRk1GT0ZRRlMvRldRL3RoaWNrbWF0aHNwYWNlRicvRlpGXHRGZW5GaG4tRiM2JkYrLUYjNiQtRj82MFEqJnVtaW51czA7RidGQkZERkdGSUZLRk1GT0ZRL0ZUUSdwcmVmaXhGJ0ZWL0ZaUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYnRmVuRmhuLUZgcjYkUSIzRidGQi1GPzYwUSMuLkYnRkJGREZHRklGS0ZNRk9GUS9GVFEocG9zdGZpeEYnRmZwRllGZW5GaG5GXXNGK0ZicS1GIzYmRl9xRmhzLUYjNiZGKy1GIzYkRmJ0LUYsNiVRI1BpRidGW3BGQkZcdUZndUYrRmJxLUYjNiYtRiw2JVEldmlld0YnRi9GMkZocy1GIzYmRitGYHRGXHVGaXRGK0YrRkJGKy1GPzYwUSI7RidGQkZERmVxRklGS0ZNRk9GUUZTRlZGXXRGZW5GaG5GKw==

11) Plot the functions NiMvJSJ6RywmKiQpJSJ4RyIiIyIiIkYqKiZGKUYqKiQpJSJ5R0YpRipGKkYq and NiMvJSJ6RywoKiYiIiMiIiIsJiUieEdGKEYoISIiRihGKComIiIlRigsJiUieUdGKEYoRitGKEYoIiIkRig= on the same axes. Plot over the rectangle with x and y between -1 and 3. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn 12) Regraph the same pair of functions, this time constraining the z-range between -1 and 7. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn These 2 surfaces have a simple relationship. Describe the relationship and replot the graph viewed at angles that make the relationship clear visually. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn The relationship is....... Viewing angles that makes this obvious are theta = ....... and phi = ........

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn