The connection is that expressions are often used to define functions. That is, we could let , which defines a function . The rule for this function is to substitute a value for into the expression to obtain the output value. Not all expressions can be used to define functions, however, and not all functions are defined by expressions so these really are distinct mathematical objects.

Maple mimics this mathematical distinction between expression and function. You can define expressions in Maple and even label them for later use with commands like the one below.

> p := x^2+sin(3*x);This is an expression not a function, which means there is no rule associated with it. Thus evaluating the expression at a specific value of requires the

> subs(x=2,p);The syntax for defining a function in Maple uses an arrow to make the idea of a function as a process explicit. For example, we can define a function in Maple using the expression with the following command.

> f := x -> x^2+sin(3*x);Evaluating our function at a specific value of is now easy.

> f(2);One final thing to note is that Maple will use

- The
`D`operator acts on a function to produce the derivative of that function. - The
`diff`command acts on an expression and differentiates that expression with respect to a variable specified by the user.

When you use the `D` operator to compute the derivative of a
function, the result is also a function, as shown below.

> D(f);If you provide a label, then you get a function you can use later in the session,

> df := D(f);However, this is usually not necessary. See the examples below.

If you want to evaluate the derivative at a specific value of or
just get the expression for the derivative, you can use the following
forms of the `D` operator.

> D(f)(2); > D(f)(x);This last form is the one to use for plotting, as shown below.

> plot(D(f)(x),x=-2..2);

Suppose you want to find the equaton of the line tangent to the graph of at the point . This can be done in Maple using the point slope form of a line as shown below.

> tanline := D(f)(5)*(x-5)+f(5);

The `D` operator cannot be used on expressions, for example
trying to use it to differentiate the expresssion we defined above
results in an error.

> D(p);If you recall that Maple uses

> D(f(x));

To differentiate expressions, you need to use the `diff`
command. Here is an example.

> diff(p,x);The

> diff(f(x),x);Note, however, that the result of the

- Compute the derivative of the following function using
the
`diff`command. Plot the original function and its derivative on the same graph over the range . In your writeup, identify which curve is the function and which is the derivative, for example by identifying a point where the derivative is zero.

- Compute the derivative of the following function using
the
`D`operator. Also, find the tangent line at and plot it on the same graph. Use the interval .

- If an object is dropped off the edge of a meter cliff and air
resistance is neglected, the position of the object at time
is given by
, where is in seconds. The
average velocity at time is

Show, using a plot or algebra that the average velocity at time is exactly half the instantaneous velocity at time .

2003-11-18