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Calculus
Computer Lab
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Derive BasicsBy |
| Expression/Action | Type: | Menu: | ||
| e | Alt-e or #e | |||
 |
Alt-p or pi | |||
 ,  |
inf, -inf | |||
The square sign:  |
alt-q | |||
 ,  |
ln x , log(b, x) | |||
| Inverse trigonometric functions | asin x, atan x, etc. | |||
 |
dif(f(x), x) | Calc/Diff | ||
 |
dif(f(x), x, n) | Calc/Diff | ||
 |
int(f(x), x) | Calc/Int | ||
 |
int(f(x), x, a, b) | Calc/Int | ||
| Simplify an expression | s | Simplify | ||
| Approximate | x | Approximate | ||
| Cancel a menu choice | Esc-key | |||
| Move around in a menu | Tab-key | |||
| Change highlighted expression | up, down arrow keys | |||
| Insert highlighted expression | F3 , F4 with ( )'s |
Supposed you wanted to graph the function x sin x. You would first
Author this and then choose Plot. You then get the submenu:
Beside, Under, Overlay. You will usually want Beside. After choosing this (by pressing the b--key or
pressing the Enter key), you are asked for the column. You can
press Enter to get column 40. This will split the screen into
two windows: an Algebra window on the left and a Plotting window on the
right. These windows each have a number in their upper left
hand corner. You
can tell which window you are in by which number is highlighted. You
can switch windows by pressing the F1-key or choosing
Algebra when you are in the plot window or choosing Plot from the
algebra window.
After you have created the plot window, you are in that window. You
need to choose Plot from that window to actually do the
plotting. This will plot the expression highlighted in the algebra
window. You can plot several functions in the same plot window.
Move to the algebra window, use the up and down arrows to highlight
the expression you want to plot, switch to the plot window (by
pressing F1 or choosing Plot), and then choose Plot from
the plot window. Now both expressions will be graphed. You can plot
as many as you want this way. The plot window also has a Delete option for removing some or all of the expressions to be
plotted.
When you plot, there is a small crosshair in the plot window, initially at the (1,1) position. You can move it around using the arrow keys. The coordinates of the position of the cross are give at the bottom of the screen. This is useful for such things as finding the coordinates of a maximum or a minimum, or where two graphs meet. The Center option will redraw the graph so that the cross is in the center of the window. You can use the Zoom option to move in or out.
We mentioned above how to plot any number of graphs simultaneously by
repeatedly switching between the algebra window and the graphics
window. Another technique for plotting three or more functions is to
plot a vector of functions. This just means Authoring a
collection of functions, separated by commas and surrounded by
brackets.
For example, Plotting the expression [x, x^2,
x^3] will graph the three functions: x, x², and x³. In
order to plot a collection of individual points one enters the points
as a matrix, for example Authoring the expression
[[0,0], [1,1], [2,0]] and then Plotting it will graph
the 3 points: (0,0), (1,1) and (2,1). In the graphics window
choose Option}/State then press the Tab key
followed by Connected. Then choosing Plot again will
graph the 3 points above but also draw the line segment between them.
See the Figure 4 where each of these techniques is
demonstrated. The color of a plot is controlled by choosing
Option/Color/Plot and then making sections
on the menu.
Click 2D-Plot to see a summary of these of these commands or choose 2D-Plotting for the online Help menu.
Figure 5. Using Plot for graphics
The main tools for manipulating the view of your graph are:
| Effect: | Type: | Menu: |
| Switch windows | F1 | Algebra or Plot |
| Zoom in, zoom out | F9, F10 | Zoom |
| Move Crosshair | 4 arrow keys | Move |
| Move crosshair quickly |
Ctrl-  , Ctrl-  , PgUp,
PgDn |
Move |
| Center on crosshair | c | Center |
| change the scale | s | Scale |
If you Author f(x), DERIVE will put f x on the screen because it thinks both x and f are variables. If you wish to define f(x)=x² + 2x + 1 for example, you could Author f(x) := x^2 + 2x + 1, or you could choose Declare from the main menu at the bottom, then choose Function, and then fill in the information asked for. See Figure 4.
Figure 4. Examples of Declare, Simplify and ApproX
Constant are treated just like functions except there are no
arguments. In order to set a = 2
for example you type a
:= 2 pi. Then, whenever you simplify an expression containing a,
each occurrence is replaced with 2. In many problems you find
it useful to have constant names with more than one letter or symbol,
which is the default in DERIVE. For example variables with names like
x1, y2, etc. will be used frequently as our names like
``gravity". This can be done by changing to word input mode by
choosing Options/Input/Word. In this mode variables can
have several letters but when in word input mode you have to be more
careful with spaces: to get 
you should enter a x^2,
not ax^2 (otherwise 
will be treated as
a variable). DERIVE indicates multiplication with a centered dot. So
on the screen you should see 
, not .
An interesting function defining technique is provided by the factorials. For n = 1, 2, ... we define n-factorial, denoted by n!, as
and for completeness we define 0! = 1. These numbers are important in many formulas, e.g., the binomial theorem. One observes the important recursive relationship n! = n (n-1)! which gives the value of n! in terms of the previous one (n-1)!. Thus, since 5! = 120 we see immediately that 6!= 720 without multiplying all 6 numbers together.
In DERIVE we can recursively define a function F(n) satisfying F(n)=n! by simply typing
F(n) := IF(n=0, 1, n F(n-1))
!
We will give several other examples of this technique in the text.
In calculus functions are typically described by giving a formula
like 
but another technique is to describe the values
restricted to certain intervals or with different formulas on different
ranges of x-values. As an example, consider the function
which defines a unique value f(x) for each value of x. The problem is how do we define such a function using DERIVE?
One basic technique is to use the logical IF statement. The syntax is IF(test, true, false). For example, if we enter and simplify IF(1 < 2, 0, 1) we get 0 whereas IF(1 = 2, 0, 1) simplifies to 1. Now our function above is entered as:
f(x):= IF(x < 1, 2x + 1, IF(x <= 2, x^2, 4))
Notice how we use nested IF statements to deal with the three
conditions and that with four conditions even more nesting would be
required. Now once f(x) has been defined we can make computations such
as Simplifying F(1) (should get 1), computing limits such
as the right-hand limit 
(should get x² evaluated
at x=1) or definite integrals using approX to simplify. We can
also plot f(x) in the usual manner described in the previous section.
Notice from Figure 0.5 that the function y=f(x) is continuous
at all 
. At x=1, both left and right limits exist but they
are not equal so the graph has a jump discontinuity.
As the number of table entries increases we are forced into using nested IF statements and the formulas become quite difficult to read and understand. An alternate approach is to use the DERIVE function CHI(a,x,b) which is simply
Then except for x = 1 our function f(x) above satisfies:
F(x):=(2x+1) CHI(-inf,x,1) + x^2 CHI(1,x,2) + 4 CHI(2,x,inf)
Vectors are quite useful in DERIVE, even for calculus. The next section shows how they are used in plotting. To enter the 3 element vector with entries a, b, and c, Author [a, b, c]. It is important to note the square brackets which are used in DERIVE for vectors; commas are used to separate the elements.
DERIVE also provides a useful function for constructing vectors. The vector function is a good way to make lists and tables in DERIVE. For example, if you Author vector(n^2, n, 1, 3), it will Simplify to [1, 4, 9]. The form of the vector function is vector(u,i,k,m) where u is an expression containing i. This will produce the vector [u(k),u(k+1),...,u(m)]. You can also use the Calculus/Vector menu option to create a vector. So, for example, to obtain the same vector as before, you Author n^2 and highlight this. Now choose Calculus/Vector and setting Variable: n (not x), Start: 1 and End: 3.
A table (or matrix) can be produced by making a vector with vector entries. If we modify the previous example slightly by replacing the expression n^2 with [n, n^2] and then repeating the above we get [[1, 1], [2, 4], [3, 9]] which displays as a table with the first column containing the value of the index n and the second column containing the value of the expression n². This is a good technique for studying patterns in data. See Figure 6 for some examples. Click VECTOR for more information on this function.
Figure 6. Using the Calculus/Vector command
We have already seen two important applications of vectors in Section 0.10; namely,
plots each of these functions in order.
will plot the individual points 
, 
.
We will have other application that will require us to refer to the
individual expression inside of a vector. This is done with the DERIVE
SUB function (which is short for subscript). Thus, for
example, [a,b,c] SUB 2 simplifies to the second element b.
DERIVE will display this as 
which explains the name.
For a matrix or vector of vectors then double subscripting is used so
that, for example, if
then Authoring y SUB 2 SUB 1 will be displayed
as 
and simplify to 3 (because it's on row 2 and column 1).
You can save your expressions to a floppy disk or the hard drive H: and come back later to continue working on them. To do this, put a floppy in say the A: drive and, from the algebra window, choose Transfer, then choose Save, then Derive and then enter a file name such as A:LAB5 or A:LAB5.MTH or H:LAB5 for the hard drive. If you don't type the extension .MTH; DERIVE will add it anyway. Later, you can reenter these expressions by starting DERIVE and choosing Transfer, then Load, then Derive, and then entering the file name A:LAB5 or A:LAB5.MTH. If you forget the name of your files just type either A: or H: and press the F1-key to select from a listing of your files.
During the course of your session with the computer you will make lots of typing and mathematical mistakes. Before saving your work to a file or before printing and turning your lab in for grading you should erase the unneed entries and clean up the file. You do this with the Remove and the moVe commands. You should practice these commands on some scratch work to make certain you understand them. There is also an Unremove command for correcting mistakes. One way to use the moVe command is to write comments in the file and placing them before computation. Many of the *.MTH files that we wrote for this lab manual use this technique. To do it, just Author a line of text enclosed in double quotes, for example, "Now substitute x=0.".
You can print all the expressions in the algebra window (even the ones you can't see) or you can do a screen dump which will print the whole screen including your graphs. If you are working at home, be sure you to first configure your printer (this is not necessary in the Bilger labs). Choose Transfer, then Print, then Option. You will probably want to choose Some for the range, Extended for the character set and Dotmatrix for the printer. The next menu you can ignore unless you want sideways printout (Landscape), just press Enter. Now choose Printer, then choose either Expressions (to print the expressions), or Screen or Window. Printing the screen or window is quite slow, so only do it when you want to include a graph.
Usually you just want to print a graph. To print just a window with a graph in it, make that its the current window, and then press Shift-F9. Typically, students turn in the labs by hand writing most of the exercise answers and then including some graphs using this method. Another technique for longer labs is to make a file as discussed above and then printing the file. Some combination of hand writing and printouts should be the most efficient.
You can obtain on-line help by choosing Help. This help feature provides information on all DERIVE functions and symbols. Suppose that you want to know how to enter the second derivative of a functions f(x) by typing. For example, maybe this expression is to be used as part of another function. There are three techniques for learning how to do this.
The first method is to use the menus with Calculus/Differentiate to enter the second derivative by typing 2 after the Order entry. Then, press Author followed by the pull-down key F3 which will enter DERIVE's way of typing the expression, in this case it's DIF(g(x),x,2). The second method is to use the online help, either by choosing Help on the main menu or pressing F1 while authoring an expression. On then, selects F (for functions) and then by pressing Enter several times one finds the appropriate page of explanations.
The third method is use our Web page, see the index at the top of this file, which also is linked to DERIVE's online help file. The advantage of this last method is that the relevant page, once found, can be kept available for further consultation. One just flips between the DERIVE window and the help window by pressing the Alt-Tab or Alt-Esc keys on your PC.
We have included a few quick reference tables with common keys used
for entering things like 
, 
and Euler's
constant e. Table 0.2
gives a summary
of commands that can be issued from the Algebra window and
Table 0.3 gives a summary
of useful commands that can be used in the Plot window.
Here are a few common mistakes that everyone makes, including the authors, every once in a while. It just takes practice and discipline to avoid these problems, although, it is human nature to blame the computer for your own mistakes. Fortunately, the computer never takes insults personally and it never takes revenge by creating sticky keys, erasing files, locking up, or anything else like that ... or does it?
and instead
I got a picture of some surface. What did I do wrong? Same problem as
above, except now DERIVE is plotting a surface z=f(x,c). You probably
want to enter a vector of functions such as
VECTOR(x^2 + c, c, 0, 4)
, 
, 
,
, and 
.
correctly, but when I
substituted x=9 and simplified I got 
. What happened?
You took the square root of a negative number which is not allowed
when you are working with the real number system. DERIVE treats this
as a computation with complex numbers and uses the complex number i
(where 
).
and
I got x=2 which I guessed from the graph but the other two solutions
were 
and 
. Where do these last two come from?
If you Factor the cubic instead of using soLve you would get
. The complex solutions come from that quadratic term.
In calculus, we just ignore those complex solutions. For example, in
approximate mode (use the Option/Precision menu) solving the
above cubic will give only real solutions.
and I got 
, what's wrong?
Nothing, DERIVE is treating the letter e as an ordinary symbol like
a or b. You probably wanted Euler's constant e which is entered
with Alt-e or #e.
instead. What's wrong?
DERIVE recognized the tan x part but treated the other symbols
as individual constants. Use atan x.
by typing [v1,v2,v3]
and I got 
instead. What happened?
You must use word input mode in order that v1 is treated
as a single variable. To do this choose Option/Input and select Word mode.
or maybe 
is defined in the file on Newton approximation.
You can check on this by scrolling up to find the definition or else
if you are sure that neither definition is needed you can select Transfer/Clear and then choose Functions. This will clear
all function definitions. If instead, you know the problem is that
x(t) is defined and you want to erase just that definition, then Author x:=.
and I got
SIN([2 ]) instead, what happened? You Author
sin[2pi] instead of sin(2pi). DERIVE treats square
brackets not as parenthesis but as a device for defining vectors,
see Section 0.13.
,
instead DERIVE returns a question mark indicating that it can't do
this problem. What's wrong? Same as above, check your parenthesis.
This last example is a little tricky because DERIVE uses square
brackets to display some expressions, when in fact, those expressions
must be entered with parenthesis.
8 will be displayed. If you Simplify this, you get 2
where
m is any integer. If we use DERIVE to solve this equation it gives
the 3 solutions corresponding to m=-1,0,1 (these are the principle
solutions and all others are obtained by adding or subtracting multiples
of 
).
cannot be solved exactly
in DERIVE although it is obvious that x=0 is one solution and by
viewing the graph we see another one with 
. In order to
approximate this solution we need to choose Option/Precision
and then set Mode to Approximate (by pressing the a-key).
Then, when we choose Solve we are asked for a range of x's
(initially it is the interval [-10,10]). Since we have (at least) 2
solutions we should restrict the interval to say [.5,1] which seems
reasonable based on the graphical evidence. The result is that DERIVE
gives the solution x=.876626. We will discuss how this computation
is done later in Chapter 5.
and you want to
evaluate this with x=3 or if you solved an equation f(x)=0 and want to substitute in that value of x, you Author the expression
and then choose Manage/Substitute. This will ask you for
the expression. It will guess the highlighted expression, which is
usually what you want so you can just hit return in this case. It
then gives the name of a variable occurring in the expression. In
this example x is the only variable. You then type over x with
the value you want to substitute, in this case 3. You can then
Simplify or approXimate. You do not have to substitute a
number for x; you can substitute another expression.
-1. Then Simplify to
get the answer. In a similar manner DERIVE does summation and
product problems. Special notations are used; namely,
, is read as ``the sum of
as i runs from 1 to n.''
Often 
to get the formula:
