Mathematics 133 – Intensive Calculus for Science 1 Lab 1: Getting Started with Maple, Plotting Functions September 20, 2005

Goals

Today, we will begin using a program called Maple to draw graphs of functions of one variable (given by symbolic formulas). You will be using the commands covered here in the other lab days for the rest of the year. On the mathematical side, we will also explore some properties of polynomial and exponential functions.

Lab Activity 1

Let’s get right down to work and walk through a sample graphing session! First, you will need to get into Windows and Maple as described in the General Information on Maple handout. The basic Maple command for 2D plotting graphs of the form y = f(x) is called plot. The basic format is:

                       plot(function,range,options);

where

1. function is the function to be plotted – the simplest way to specify one is via a formula (an expression in Maple)

2. range is the range of x-values you want to see plotted, and

3. options can be used to control the form of the plot if desired. No options need be specified, however, so that part can be absent. More on possible options later.

The above plot command is a “template.” To do an actual plot, you will need to fill in the expression for f(x) and range of x values for the plot you want. For example, suppose we wanted to plot y = x^{3} + 5x^{2} + 3x − 5 for x between -3 and 3. We could use the Maple plot command with no options:

                       plot(x^3+5x^2+2*x-5,x=-3..3);

(Note: The close parenthesis comes directly after the range of x-values if there are no options.) Type in this command line (exactly as here) and press ENTER. If you make a typing mistake, Maple will let you know about it(!) Fortunately, if this happens, the whole command does NOT need to be re-entered. Just move the cursor arrow to the place on the input line you want to change, press the left mouse button, and edit the input as needed. Typing from the keyboard will insert new stuff at the cursor location; the DELETE and BACKSPACE keys will remove stuff (DELETE removes the character in front of the ”insert point”; BACKSPACE removes the character in back). You can also move around on the input line with the arrow keys if more than one thing needs to be changed. When you think it’s OK, press ENTER again to have Maple execute the command again.

When all goes well you will see the graphics output displayed under the input command in the worksheet.

From the formula, you might guess that there is at least one other x-axis intercept for this graph (why, where?). To see more of the graph as well, edit your previous command line to change the interval of x values (do not retype the whole command). Press ENTER on that input line to have Maple execute the command again. Experiment until you are sure that your plot shows all the x-intercepts of this graph. (You can repeat this process of editing a command and re-running it as often as you want; the previous output is replaced by the new output each time.)

You can also manipulate the graphics output in place within the worksheet in several ways. For instance:

• If you click the left mouse button once over the graphics output, you will see a black box with eight “tabs” displayed at the corners and the midpoints of the edges of the box. If you place the cursor on one of the tabs, hold down the left mouse button, drag the cursor and release, you can resize the graphics. Try it! Note: This feature is especially useful when you go to print your worksheet. The ”default” size for printed versions of plots is so large that a single plot will take almost one whole sheet of paper(!) Please save trees by resizing your plots to smaller sizes(!).

• Maple 2D graphics output regions have another nice feature: If you place the cursor arrow at a point in a 2D graphics region and click the LEFT mouse button once, approximate coordinates of the point at the head of the arrow are printed out in a box at the upper left of the window. How good approximations can you get that way to the x-intercepts of this graph?

• Other features of the graph can be changed from the tool bar. Experiment and see if you can figure out what the different tool bar icons do.

When you get a graphics window with all the intercepts shown, let’s add two more things.

• First, it’s possible to show more than one graph on the same set of axes. To do this, you place the formulas for the functions you want in a set of square brackets ([ ]), separated by commas. In the options section of the plot command, you can then specify which colors you want to use to make the graph more informative. For example, the option

                         color=[black,red]

will color the first function you listed black, and the second one red. Most other common colors are available too, by their usual names, and you can “mix and match” at will. (Unfortunately we don’t have a color printer in the lab, but at least you will see the colors on the monitor). In your plot, add a second graph, y = 10 sin(x) and color the two plots blue and red.

• Sometimes, an informative title makes a graph much more understandable. To add a text title to a plot, you can insert a comma after the range of x-values, and include an option in the plot command of the form

                     title="whatever you want"

Add an appropriate title to your plot.

Include the final graph you generate here in the worksheet you submit for this assignment.

More Detailed Information on Maple Commands

Now that we have seen some first examples of Maple commands, here is some more information about the syntax rules that Maple uses to decide if what you have typed in is a well-formed command it can execute.

For a function described by a formula, the formula is entered in something like usual mathematical notation:

• The symbols for addition, subtraction, multiplication, and division are +, -, *, / respectively.

• The caret (ˆ ) is the Maple symbol for raising to a power.

• The asterisk symbol for multiplication MUST be included whenever you are performing a product in a formula. Moreover, everything must be entered in one string of characters, so you will need to use parentheses to group terms to get the expressions you want.

• Maple “knows” all the usual functions from the “library” in Chapter 1 of our textbook. The names of the most common ones are sin, cos, tan, exp, ln. To use one of these functions in a Maple formula, you put the name, followed by the “argument” (that is the expression you are applying the function to) in parentheses.

• The range of x-values to plot for a graph y = f(x) MUST be specified, in the format x = lowx..highx. A range of y-values to plot CAN also be specified, in the format

y = lowy..highy.

• Every Maple command must be terminated either with a semicolon or a colon. This is the signal that the command is finished and that Maple should try to execute what you asked it to do. The difference between a semicolon and a colon is that for commands terminated with a semicolon, any output produced will be displayed. If you terminate the command with a colon, the calculation will be performed but not displayed (this is useful sometimes for intermediate steps in a big computation where you don’t need to see the output). Until you enter either a semicolon or a colon and press ENTER, Maple will not do anything with your command. Also, if a command you want to enter doesn’t all fit on one line, just keep typing, but don’t press ENTER until you are finished. Maple automatically wraps around to a new line if you need it.

Lab Activity 2 Questions

A) Power functions f(x) = x^{n} and exponential functions g(x) = a^{x} with a > 1 both increase as x increases. If we only plot a small range of values, though, what we see might be misleading about which function is growing faster as x → +∞.

1) For example, plot f(x) = 2^{x} (2^{x} in Maple) and g(x) = x^{3} together on the same axes for x = 0..3. Which one seems to be growing faster?

2) What happens if you extend the range of plotting to x = 0..8?

3) Your last plot should suggest that the exponential might actually be “catching up” to the power and getting ready to pass it. To see if that is true, we can use a different feature of Maple to solve the equation x^{3} = 2^{x}. The command

                     fsolve(x^3 = 2^x, x=10);

will do this (approximately), looking for x “near 10”. Take the number you get, add 3 and plot the two functions again on the range from x = 0 to that number.

4) Repeat the previous parts for f(x) = 2^{x} and g(x) = x^{4}, x^{5}, and x^{6}. (Note: You will need to change the fsolve command to include an appropriate place to look for solutions of x^{n} = 2^{x} for the different values of n.) Do any of these functions keep growing faster than f(x) = 2^{x} indefinitely? Where does each of them get overtaken by the exponential? (Note: If you are unsure about what your graphs are showing you, also read section 1.6 in the text!)

B) (Some polynomial graphs)

1) Execute the plot command:

            plot(2*x^4-20*x^3+70*x^2-100*x+48,x=0..5);

How many turning points does the graph have? (A turning point is a point where the function “turns” from being increasing to being decreasing, or vice versa.)

2) What would be the factored form of this fourth-degree polynomial? Explain.

3) The graph above shows one of the possible shapes for the graph of a 4th-degree polynomial. There are others, too. We can see some of them by changing the coefficient a in the polynomial p(x) = 2x^{4} − 20x^{3} + 70x^{2} + ax + 48. (The graph from part 1 has a = −100. Find a value of a for which the polynomial a different number of turning points than the graph in part 1 above.

Assignment

The lab write-up (a paper print-out of your worksheet, including all graphs and answers/explanations in text regions) is due on Monday, September 26. One write-up for each pair of lab partners.