# Basic mathematic instructions

**Note**: There are several live demos on this page, identified with the label INTERACTIVE DEMO. If you click into the area containing the CalcMe parameters, it will launch a new browser window containing a CalcMe session with all the operations you can see. Try it out, change the parameters or options. You won't break anything.

## Graphic representation of a function

If you want to create the graph of a function, you just have to type the expression of the function you want to represent and click on the *Draw* action. You can also use the keyboard shortcut `Ctrl`+`Shift`+`P`.

Then, a plot will appear with the function represented at the right-hand side of the screen. By default, the appearance of both the graph and the plot follows a default style.

If you want to change some of the look and feel, you can do it by initially defining a plot with the desired properties. For instance, you can change the center, width, height, and visibility of the grid using the given commands.

The appearance of the graph changes completely and can be adapted to your needs by introducing these changes.

INTERACTIVE DEMO

## Graphic representation of a set of functions

As with a function, it's possible to represent a set of functions on the same plotter. You just have to write each function in a different line and perform the corresponding action.

INTERACTIVE DEMO

In order to distinguish, beyond the color, what function each representation corresponds to, you can set its label so that it appears permanently.

In fact, you can also change the color, thickness, and line style by using the buttons right there. Try it and you will see all the available options!

## Limits

If you want to calculate the limit of a function, there are mainly two ways to do it. You can use the command `limit`

or the icon you can find in the *Calculus* section of the *Menu*, being this second option quite common.

In fact, using the action you can easily calculate limits and side limits (both right and left) by just indicating the function and the point. Please note if you want to calculate limits at infinity, you must enter it using the icon you can find in the *Symbols* section of the *Menu*.

INTERACTIVE DEMO

These limits calculations can help us, for instance, when we look for asymptotes in a function. Given the function , we want to check that it has no horizontal asymptotes and that it has an oblique asymptote of the form . Pay attention to the graphical representation as our function increasingly approaches its oblique asymptote.

INTERACTIVE DEMO

## Domain

Given any function, you can easily obtain its domain using the `domain`

command. This will return the set of values for which the function is defined.

INTERACTIVE DEMO

Similarly, given a function and any point, you can check if such a point belongs to the domain of the function by using the command `belongs_to_domain?`

. Note it's necessary to specify the variable we are referring to.

INTERACTIVE DEMO

## Derivatives

Given any function, you can find its derivative in three different ways: using the action you can see in the top bar, using the `differentiate`

command, or using the icon you can find in the *Calculus* section of the *Menu*. This third option, shown in the screenshot, is the most common.

In fact, using the icon you only have to write the function to be derived in the numerator and the variable with which you want to calculate the derivative in the denominator. Furthermore, you can easily change it in case the function is multivariate.

INTERACTIVE DEMO

On the other hand, the aforementioned `differentiate`

command allows, among others, to directly find the nth derivative of a function.

INTERACTIVE DEMO

## Integrals

As with derivatives, given any function, you can find its integral by three different procedures: using the action you can find in the upper bar, through the `integral`

command, or using the icon present in the *Calculus* section of the *Menu*. The third option, shown in the screenshot, is the most common.

In fact, using the icon you will be able to calculate both definite and indefinite integrals just by indicating the function and, if it's the case, the integration interval. Note if you want to calculate improper integrals at infinity, you will have to enter them using the icon you can find in the *Symbols* section of the *Menu*.

INTERACTIVE DEMO

On the other hand, you can also calculate multiple integrals by concatenating simple integrals. Be careful with the order in which you write the differentials, you can get different values!

INTERACTIVE DEMO