Mathematics 133 -- Intensive Calculus for Science

Lab 1: Getting Started with Maple, Plotting Functions

September 25 and 26, 2000

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 for most of the other lab days this semester. On the mathematical side, we will study another graphical way to understand the "zooming in" operation involved in finding the instantaneous rate of change of a function.

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 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 if you don't want to. More on this later.

The above plot command is a template. To do an actual plot, you will need to fill in the expression and range of x values for the plot you want and add the appropriate options. For example, suppose we wanted to plot y = x4 - 2x3 + x - 5 for x between -2 and 1. We could use the Maple plot command with no options:

plot(x^4-2*x^3+x-5,x=-2..1);

(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-intercept for this graph for x > 1 (why?). To see that part of the graph as well, edit your previous command line to change the right hand endpoint of 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 display several plots together by putting the formulas for the functions together inside the plot command, in a set of curly brackets ( { } ), separated by commas. For example, try entering the command

plot({x^4-2*x^3+x-5,cos(x)},x=-2..1);

to plot our polynomial function from before together with y = cos(x).

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

  1. 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!
  2. 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?
  3. 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 one more thing. 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`

NOTE: Those are "backquotes," not ordinary apostrophes. Add an appropriate title to your plot.

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:

  1. The symbols for addition, subtraction, multiplication, and division are +, -, *, / respectively.
  2. The caret (^) is the Maple symbol for raising to a power.
  3. 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. The rule to keep in mind is: Maple always evaluates expressions by doing powers first, then products and quotients, then sums and products, left to right, unless parentheses are used to override these built-in rules. For example, the Maple expression a + b^2/c + d is the same as the mathematical formula: a + b2/c + d. If you really wanted a + b2 in the numerator and c + d in the denominator of a fraction, you will need to enter the expression (a + b^2)/(c + d). What if you really wanted (a+b)2 in the numerator?
  4. Maple ``knows'' all the functions we studied in Chapter 1 of our text. 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.
  5. The range of x-values to plot for a graph y = f(x) MUST be specified, in the format x = low..high. A range of y-values to plot can also be specified, in the same format.
  6. 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

Recall, last Monday in class, we introduced rates of change of functions. The average rate of change of a function f(x) over the interval [a,b] in x is (f(b) - f(a))/(b - a). We can attempt to measure the "instantaneous" rate of change of a function f at x = a by studying (f(a+h) - f(a))/((a + h) - a)} and letting h approach zero. Today, we will use Maple graphing to understand the meaning of this process in another, geometrical way.

Questions

In these questions, we will study the graph

y = f(x) = sin(x) - x cos(x).

To plot this function in Maple, you will need to translate the mathematical formula into a Maple expression.

  1. Plot the portion of the graph y = f(x), for -5 <= x <= 5.
  2. Our function f(x) satisfies f(-1) = -.3012 (approximate value). We want to use the graph to find an instantaneous rate of change at x = -1. The process of computing average rates of change over shorter and shorter intervals can be seen visually if we "zoom in" on x = -1. Here "zooming in" will mean plotting smaller and smaller pieces of the graph taking x in smaller and smaller intervals containing x = -1. For instance, try plotting the function on the new intervals -1.5 <= x <= -.5, then -1.05 <= x <= -.95, then -1.005 <= x <= -.995. What are you noticing? To be sure this is a real pattern, try adding the option scaling = CONSTRAINED in the plot command from now on. (This plots with equal scales on the x and y axes.) What do you see as we ``zoom in'' this way? Keep only your final graph in the worksheet (the others are not necessary -- and long worksheets take longer to print out).
  3. A "zoomed" graph like the one on the interval [-1.005, -.995] can be used to compute an approximation to the instantaneous rate of change at x = -1. If you place the cursor over the graph in your Maple worksheet and click the left mouse button, the approximate coordinates of the point at the head of the cursor arrow will be displayed in a box under the toolbar in the Maple V Release 5 window. Clicking on two different points will give you the information you need to compute a slope. Maple can also be used as a numerical calculator if you want to compute the slope value. For example, try entering a Maple command like (6.2-5.3)/(.443-.323); When you press ENTER the value will be computed and displayed. Use these ideas to find an approximate value for the instantaneous rate of change of f at x = -1.
  4. Also find the equation of the line through (-1,f(-1)) whose slope is your instantaneous rate of change, and plot it together with f(x) for x on the interval -5..5.
  5. Repeat the ``zooming process'' near the point (4,f(4)), describe what happens, include your final "zoomed" graph in your worksheet, and use it to approximate instantaneous rate of change
  6. Find the (approximate) equation of the line through (4,f(4)) with slope equal to your instantaneous rate of change value, and plot it together with f(x) for x on the interval x=-5..5 and take y=-5..5 to control the "plotting window".

Assignment

The lab write-up (a paper print-out of your worksheet, including all graphs and explanations in text regions) is due on Tuesday, October 3.