Mathematics 136, Section 4 -- AP Calculus

Lab 1: Getting Started with Maple, Plotting Functions

September 15 and 16, 2003

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 limit operation involved in the definition of the derivative of a function and what the property of differentiability means.

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 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 - x³ + 2x - 5 sin(x²) for x between 1 and 3. We could use the Maple plot command with no options:

plot(x^4-x^3+2*x-5*sin(x^2),x=1..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-intercept for this graph (why?). 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:

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! 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(!) {\it Please save trees} by resizing your plots to smaller sizes(!).}
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"

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:

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 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.
5. 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.
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 week in class, we introduced the derivative of a function at a point by considering instantaneous rates of change. We can attempt to measure the instantaneous rate of change of a function f at x = a by studying lim_h->0 (f(a+h) - f(a))/h} If there is a unique, finite limiting value as h ->0 (from both the positive and negative directions), we say f is differentiable at x = a and we call the number obtained in the limit f'(a), the derivative of f at a.

Also, if f is differentiable at x = a, then the tangent line at that point is the line with equation

(it's the line through (a,f(a)) with slope f'(a)). Today, we will use Maple graphing to understand the meaning of the differentiability property for a function at a point in another, geometrical way.

Questions

• A) In this question, we will study the graph
  
  y = f(x) = cos(e^x/2 - x⁴).

  To plot this function in Maple, first define a Maple function with the following command:

  f := x->cos(exp(x/2)-x^4);

  1. Plot the portion of the graph y = f(x), for -2 <= x <= 2 as follows:

     plot(f(x),x=-2..2);

     (Note: f(x) is the expression (formula) defining our function.)
  2. Now enter

     plot((f(-1+h)-f(-1))/h,h=-1..1);

     and repeat with smaller and smaller ranges of h-values: h=-0.1..0.1, h=-0.01..0.01, h=-0.001..0.001.

  3. What do your plots suggest about lim_h->0 (f(-1+h) - f(-1))/h? Estimate the number f'(-1) by interpreting your plots, and explain what you are doing in a text region.
  4. Next, we will generate a different plot showing the graph y = f(x) together with the tangent line at x = -1 and ``zoom in'' toward x = -1. To plot more than one function on the same set of axes, you can put the formulas into a list in the plot command, like this: plot([fn1,fn2],x=xlow..xhigh);. Using your estimated value for f'(-1), plot our f(x) together with the linear function f(-1) + f'(-1)*(x+1) defining the tangent line. Make your first plot for x in the interval x = -2..0. Then ``zoom in'' by reducing the range of x-values to -1.1..-0.9, -1.01..-0.99, and -1.001..-0.999. 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.) Describe in words what you are seeing, in your own words, in complete sentences.
• B) In this question, we will look at a new graph:
  
  y = g(x) = |x - 2|^1/5 cos(x²)

  (In Maple format, this function is: abs(x-2) ^ (1/5)*cos(x^2) (abs is the absolute value function.)

  1. Repeat the parts 1, 2, 3 of question A for this new function, taking a = 2, rather than a = -1. How is this case different? Does g'(2) exist?
  2. Do the same sort of zooming process on the graph y=g(x) as in part 4 of question A, but on the graph of g alone (no tangent line). What do you see here?
• C) The behavior you have seen in parts A and B is actually representative of what happens when a function is differentiable at some x = a, and what happens when a function is continuous but not differentiable. (Which function was which?) In your own words, summarize the differences between these two examples. In terms of the zooming processes we have used here, what does it mean to say that a function is differentiable at x = a?

Assignment

The lab write-up (a paper print-out of your worksheet, including all graphs and explanations in text regions) is due on Wednesday, September 17. One write-up for each pair of lab partners.