Special Menu

The following page describes the less obvious buttons on the special menu.

Wolframα [WA]

Takes the current stack entry and sends it to Wolfram Alpha online for detailed symbolic analysis. Among other things, Wolfram Alpha will calculate derivatives, integrals, roots, and factored forms of formulas. If you haven't used Wolfram Alpha before, it is highly recommended you try it out.

( )

Works in a similar way to the "{}" button in the general menu, except it is intended to designate an ordered list (such as an x,y pair when doing statistics). In general the "{}" buttons should be used most of the time except when designating an ordered pair.

(x,y) [OrderedPair]

This is a quick entry button for turning the last two entries on the stack into an ordered list (see the statistics menu for more information). It saves a couple of button clicks over using the "()" buttons.

∑{...} [Sum]

Sums the members of a list, returning a single value.

=

Exists only to create equations for analysis by Wolfram Alpha. Otherwise does nothing.

f( )→(...,f) [Coords]

Returns the inputs plus the result from the previous operation as an ordered list. For instance, if the stack contains:

    3²

This operation will return (3, 9). This operation was intended mostly as an aid to generating tables of values for graphing, and is most powerful when used in conjunction with lists. For instance, if the following entry is on the stack:

    {1, 2, 3, 4}²

This operation will have the following result:

    {(1, 1), (2, 4), (3, 9), (4, 16)}

The "Graph" button may then be used to graph these points on an x-y axis.

f' [Derivative]

Numerically estimates the value of a derivative at a given input value. The stack inputs are:

function to differentiate, value at which to find derivative

For instance, after entering:

x sin Pi 6 / Derivative

The stack will say "derivative(sin(x), π/6)" which means the derivative of sin(x) at π/6 which should show a result of 0.5. Note that derivatives of trig functions should always be done in radians.

Calculating Higher Derivatives and Graphing Derivatives:

It is possible to find a second derivative (or higher) by using a variable as the value for which the first derivative is to be found (i.e., enter 'x' or some other variable as the second value on the stack), then taking the derivative again at a numerical value. For instance, entering:

x sin x Derivative Pi 6 / Derivative

Will first result in a formula for the derivative of sin(x) at x, then a derivative of that formula will be found at the x value π/6. That is, the stack will say "derivative(derivative(sin(x), x), π/6)". This will be the second derivative of sin(x) at π/6, which should be cos(π/6) = √3/2 = 0.866.

Finally, entering:

x sin x Derivative graph

Will graph the derivative of sin(x) - i.e. "graph(derivate(sin(x), x))", which will look like a cosine wave.

Note: the numerical estimation of the derivative is found using a five-point stencil with h set to .001 or x * .001, whichever is larger. This works really well for average textbook cases, but is less accurate for cases with a lot of curvature relative to the magnitude of x (i.e., "derivative(sin(x), 320π)")

∫ [Integral]

Approximates the integral of a function using numerical integration. The stack inputs are:

function to integrate, lower bound, upper bound

The numerical integration algorithm used is Boole's Rule applied over 1000 steps (requiring 4001 function evaluations). For "friendly" cases this is quite accurate, but there is no attempt made to adapt the algorithm to more pathological cases (i.e. curves with lots of sharp bends and/or any discontinuities) so place your trust accordingly.

Matrix...

The operations in this sub-menu are discussed in detail under Matrix Operations.

Const Buttons

There are two Const buttons. The first button allows the user to peruse the library of available constant values with the constants grouped into families. The second button allows the user to search the same library using a search string. Selected constant values are placed on the stack the same way as constants like pi or e (i.e. as one or two letters representing a value), so that the large number of digits of accuracy don't clutter up calculations.

If there are additional families of constants you would like added to the library, contact the author at john@linstrum.net. Of course you can always save off your own constants using the mem menu.

Convert

Brings up a special menu to convert feet to meters, etc. The conversion will be performed on whatever entry is currently selected on the stack.