In this “section”, we will begin to look at the mathematical software “Maple”.

We will introduce the basics of defining functions, and then look at some calculus specific ideas.

Introduction to Maple

Beginning a New Document

Maple has many “packages” of commands that must be loaded in order for them to be available.

The ones we would use most often in a calculus class are called “plots” and “student”. So our first commands would be:

with(plots);

with(student);

Defining a Function

The command line for defining a function is shown below in generic form:

name:= input -> formula;

For example, to define we would enter:

f:=x-> x^2 + 5;

Some Special Functions

Maple has some specific ways of defining some common functions:

Graphing a Function

Having already defined a function (named f), we graph it as follows:

plot(f(x), x=xmin..xmax, y=ymin..ymax, color=__);

For example, to graph (which we already defined) in the window [-1, 2] × [0..10], we would enter:

plot(f(x), x=-1..2, y=0..10, color=magenta);

Example 1

Define the function and then produce a graph of this function, in blue, for values of x in the interval [-2, 6].

Graphing 2 Functions

Having already defined 2 functions (named f & g), we graph them as follows:

plot([f(x), g(x)], x=xmin..xmax, color=[__, __]);

Note that the y range is never needed, but can always be added as an extra parameter. This is true for any number of functions being graphed.

Example 2

Define the functions and then produce a graph of these functions, in blue and green respectively, for values of x in the interval [-3, 4].

Limits

Having already defined a function (named f )

We find as follows:Limit(f(x), x=a);

value(%);

We find as follows:Limit(f(x), x=a, right);

value(%);

We find as follows:Limit(f(x), x=a, left);

value(%);

Example 3

Find the following limits using Maple:

Derivatives

Having already defined a function (named f )

We find its derivative as follows:Diff(f(x), x);

value(%);

We find its nth derivative as follows:Diff(f(x), x$n);

value(%);

Example 4

Using Maple, define and then find:

Implicit Derivatives

To find for an implicitly defined curve, we enter:implicitdiff(equation, y, x);

For example, to find for the curve we would enter: implicitdiff(sqrt(x*y)=2*x, y, x);

Example 5

Use Maple to find a formula for for the given curves.

Integrals

Having already defined a function (named f )

We find as follows:Int(f(x), x);

value(%);

We find as follows:Diff(f(x), x=a..b);

value(%);

Example 6

Use Maple to find each of the following integrals: