Using the calculus applets
A typical calculus applet, for graphing a function, appears below.
At the top of the applet is a drop down menu of
examples. Click on it to drop down the menu, then click on one of the
selections. Note that the function definition (just below the graph)
changes, as does the graph itself. Many of the applets have an examples
menu at the top.
Enter your own functions
You can type your own function definition.
Click-drag over the current function definition and type a new one, such
as x^3, then press Enter. You will see your cubic on the graph. More
information on the syntax of function definitions can be found
Error messages show up in the graphing area. For
example, type in the function z^3 and press Enter. Since this applet
doesn't know about the variable z (only about x), it displays an error
message. To fix things, just edit the z back to an x and press Enter.
Zoom and pan
You can zoom in and out using the zoom buttons on
the Limit Control Panel, shown to the right of the graph. You can also
directly edit the xmin, xmax, ymin, and ymax fields, pressing Enter to
cause your limits to take effect (or click the Set Limits button).
Pressing the Restore Limits button returns you to the last saved set of
limits, useful if you've messed around with the limits and want to get
back where you started. Some applets also have a Save Limits button,
which saves the current limits for use by the Restore Limits
You can also zoom and pan using the mouse. Click on
the graph and the applet will zoom in by a factor of 2, centered on the
location of the click. Do this with the Shift key held down to zoom out
x2. Click-drag on the graph window creates a rectangle; when you let go
of the mouse button, the graph will be zoomed to that rectangle.
Click-drag with the right mouse button (or while holding down the command
key on a Macintosh) will pan the graph.
You can undo your zoom/pan experiments with the Restore Limits button.
Equalizing the axes
The Equalize button will cause both axes to have
the same scale (i.e., circles will look like circles instead of
ellipses). This is useful if you want the scales to be the same on both axes and you have
resized the window or zoomed in a way that the axes no longer have the
Try dragging the slider at the bottom of the
applet, and watch the crosshair move along the graph. Many applets have
sliders that allow you to adjust a value. You can also type a value into
the x = box next to the slider, and have the slider updated directly to
your desired value. Once you have clicked on the slider, you can also move the slider using the ← and → keys on your keyboard.
Full screen mode
If you want to project an applet onto a screen for
use in a lecture, the normal applet's lines and fonts may be too small.
Just below each applet is a button that opens a new window
with the applet in it, also with larger fonts and wider lines. Try
clicking the button below to see what this looks like. The resulting
window can be resized or maximized to fit your screen.
Limitations of the applets
There are times when the applets may occasionally produce wrong or unexpected results.
These conditions are explained in Graphing Issues and Errors.
The syntax for the definition of a function is very similar to that
found on common graphing calculators. Binary operators such as + (addition),
- (subtraction), * (multiplication), / (division), and ^ (exponentiation)
follow the standard rules for precedence. Operators at equal levels are
performed left to right, and grouping parentheses are also supported. The
following are the symbols and operations allowed in a function
|+ - * /
||The standard arithmetic binary operators.
Multiplication is implied if the * is left out (e.g., 2*x and 2x are
equivalent, but note that 2*3 is not the same as 23). Example:
The – sign can also be used as negation, as in – ( x + 2
). Use parentheses when an exponent has a - sign, as in e^(-1).
||Parentheses are used for grouping and also to
delimit the arguments to a function. Examples: (x-2)/3 and
||The independent variable used in all function
definitions (some applets support other variables, as noted on those
||Exponentiation binary operator. If the exponent
is not an integer, the program checks whether it is a rational
number. If the reduced denominator is even, or if the program cannot
determine that the exponent is rational, then only the non-negative
part of the domain is graphed. If it is rational and the reduced
denominator is odd, then the negative part of the domain is also
graphed. For example, x^(1/3) will graph a domain of all reals, while
x^(pi) will only use non-negative reals.
||Factorial, as in x!
||Inverse cosine (radians)
||Inverse sine (radians)
||Inverse tangent (radians)
||The smallest (closest to negative infinity)
real value that is greater than or equal to x and is equal to a
||Exponential function (i.e., e^x)
||The largest (closest to positive infinity) real
value that is less than or equal to x and is equal to a mathematical
||Natural logarithm (base e)
||Base 2 logarithm
||Common logarithm (base 10)
||The closest integer to x
||Drop any digits after the decimal
A special syntax is provided for conditional expressions,
which enables you to graph piecewise functions.
A conditional expression is an expression using the ? operator. For example
((x > 0)? x : –x)
“if x is greater than 0, then the value is x, otherwise it is
The part before the ? is the condition and compares two
quantities using one of the comparison operators =, >, <, >=, <=,
or <> (not equal). You can also write more complex expressions using
& (the AND binary operator), | (the OR binary operator), and ~ (the NOT
The part between the ? and the : is the value if the
condition is true and can be any valid expression (even another conditional
expression). The part after the : is the value if the condition is false.
Note that the parentheses surrounding the conditional expression are not
required, but are recommended if the conditional expression is part of a
The false part (after the :) is optional. If it is not
present, then when the condition is false the expression evaluates to
“not a number,” which will cause nothing to be graphed for that
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Other differentiation topics
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.