Suppose you want
to plot a simple graph such as y = x+1
From the menu,
click Graph, then Graph Expression.
Into the text
box type in x + 1 then click
OK. You will see the graph.
It is more useful to set up a function first. From the menu click Functions..Define..Then type in:
f(x) = x + 1
Typing Enter or clicking OK will define this function, and its graph will be shown.
File.. New and File..Save
Memo - this is some text saved with a file, giving an explanation or notes about
the idea behind the file contents. The memo is displayed when the file loads,
and can be reviewed/edited from the menu Memo option.
Files remember the 'context' of what was saved eg polar or parametric functions,
and the appropriate dialog box is displayed when the file opens.
Mouse right-click on the graph background offers various options, some of which are obvious - zoom in and out, centre, and reset garph limits to the original : +/- 5. The non-obvious options will display a chord on a function, showing the 'rise and run', and the slope of the chord as 'rise over run'. With appropriate zoom in, continuous functions become locally straight, and the chord slope becomes the function differential coefficient. The other options enable the user to select where (what x value) the chord is shown at, or to use the next function, if several functions are being displayed.
Two implicit relations can be displayed. This is to enable simultaneous equations to be written in conventional form.
If a mouse wheel is available, it does zoom in and out.
Some 'example' files are supplied to show what can be done.
Descartes is a program to help students of algebra and calculus visualise the ideas involved.
It displays graphs in a variety of forms - Cartesian and polar co-ordinates, parametric functions, and relations defined intrinsically. It supports a wide range of standard 'bulit-in' functions, and it is possible to define an unlimited number of other functions. These functions can have an unlimited number of arguments. Named values - called variables - can be set up, and the value of them altered. Values for e and pi are pre-defined.
The notation is similar to conventional mathematics except that -
Many more examples are given below
This provides information on the co-ordinates of the mouse pointer, in terms of Cartesian x y and polar r theta systems. In addition if the mouse is dragged over the graph, the information box displays the length and slope of the line segment :
The information box can be dragged to a convenient location, and Settings.. Show info box make it disappear and reappear.
A new user-defined function can be set up by clicking Functions..Define. In the resulting dialog box enter the function:
This example defines a function called f, with one argument, named x. Functions can have names with more than one letter (but spaces are not allowed in the function name), and there can be several arguments. For example
Graphs are usually displayed with x as the independent variable - going across the screen. Other variables can be altered, typically like this
If a function is defined with the same name as an existing function, the old version will be overwritten.
The first ten defined functions are usually displayed in graph form. This can be controlled by Function.. Select graph.. Only checked functions are displayed :
A defined function can be edited by Function.. Edit.. then clicking next to the function to edit:
The following are available:
|
Group |
Notation |
Meaning |
|
Trigonometric |
sin |
sine (all angle measure is in radians ) |
|
|
cos |
cosine |
|
|
tan |
tangent |
|
|
acos |
inverse cosine ( 0 to pi ) |
|
|
asin |
inverse sine ( -pi/2 to +pi/2) |
|
|
atan |
inverse tan ( -pi/2 to pi/2 ) |
|
|
sec |
sec or 1/cos |
|
|
cosec |
cosec or 1/sin |
|
|
cot |
cot or 1 / tan |
|
Hyperbolic |
cosh |
cosh |
|
|
sinh |
sinh |
|
|
tanh |
tanh |
|
Exponential |
log |
natural logarithm ( base e ) |
|
|
|
note exponentiation is ^ and e is predefined - so |
|
|
|
the exponential function is e ^ x |
|
Numeric |
mod |
absolute value, or |x| |
|
|
! |
factorial |
|
|
ceil |
ceiling : ceil(x) = smallest integer greater than x |
|
|
floor |
floor : floor(x) = largest integer less than x |
|
|
sqrt |
square root (positive root only) |
|
Calculus |
diff |
differential coefficient - note there are 2 arguments - for examples |
|
|
|
|
|
|
|
|
|
|
diffn |
nth order differential coefficient - 3 arguments - for example |
|
|
|
|
|
|
int |
The proper integral of the first argument, with respect to the second, from the third -- example |
 |
||
|
Series |
sum |
Sum of series. The four parameters are the index to sum over, the first and last value sof it, and an expression for each term in the series. For example: |
|
|
|
|
|
|
prod |
Product of terms. Parameters as for sum. For example Wallis' product |
|
|
|
|
|
Other |
gamma |
The gamma function |
Variables can be given values, or have them changed, by clicking Variables on the menu: In the left column variable names can be added. In the second column new values can be entered. Clicking the buttons labelled '-' and '+' will decrease or increase the value by 10% (so they will have no effect if the value is 0).
Variables can be referred to in expressions and functions - when encountered they will be set up with inital values of 0. Name restrictions are as for functions - unlimited length but no spaces.
Variables are 'global' . 'Variables' named e and pi are set up with appropriate values.
All user-defined functions ( Functions.. Define.. ) are graphed against x as the independent variable x. Selected functions can be hidden, by Functions.. Select graph. If a function cannot be seen, it may be outside the graph boundaries. Select Graph.. to alter the graph limits in various ways.
In a similar way an expression in terms of x can be graphed by Graph.. Graph expression
From the menu simply go Parametric..Setup. Then enter expressions
for x and y in terms of some 'parameter' - any other vaiable, though
t is often chosen. Enter which variable is the parameter, and the lower and
upper limits of it.
Then clicking on plot will draw the curve, clear will erase the display, and Close will close the dialog.
If you move the slider, and red cross moves over the curve, and the value is displayed, so that you can see which points on the curve correspond to which values of the parameter.
These
are graphs of functions using polar co-ordinates, r and theta. Theta is the
angle in radians anti-clockwise from the x axis, and r is the distance from
the origin. Any variable can be used to correspond to the theta variable, and
the selected function is plotted as r. Simply define the function in terms of
the angle (say t, so you might define f(t) = 2*sin(3*t) ). Then from the menu,
go Polar, and select the function you want to plot. What you get is shown here.
These are restricted to y as an implicit function of x. This means that instead of y = .. some expression in x, we have for example
or in general
f(x,y) = g(x,y)
Just choose implicit from the menu and enter the left and right hand side of the expression. For example:
This shades areas of the plane where one expression is greater than another. For example
It sometimes makes sense to combine an inequality with a plot of an intrinsic function, such as:
.gif)
This seems to show that if y is defined by
sin(x+y) = y
then y has a maximum near (0.57, 1).
Can you use calculus to work out why?
