- Purpose
- Background
- Functions and Expressions in Maple
- Something you should never do
- The Maple D and diff commands
- Tangent Lines

- Exercises

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

> F(x) := x^2+3;

Since Maple doesn't complain, students often think that what they've done is correct. The output from the following commands, however, shows that the object we've defined doesn't behave like a function.

> F(x);

> F(1);

> F(t);

> F(2*x);

What is happening here is that we've defined something called a

- 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);

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);

Error, (in D) univariate operand expectedIf you recall that Maple uses

> D(f(x));

Error, (in D) univariate operand expected

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

To find the tangent line at to our function , we
need two pieces of information. One is the slope of the tangent line,
which is given by
, and the other is a point on the
line, which is
. Then, using the
point-slope formula, the equation for the tangent line is

Implementing this using the `D` operator is relatively simple,
as shown below.

> tf := x -> D(f)(3/2)*(x-3/2)+f(3/2);

> plot({f(x),tf(x)},x=0..3);

Using the `diff` command to compute the tangent line is a
little harder, because the `subs` command must be used.

> tf2 := x -> subs(x=3/2,diff(f(x),x))*(x-3/2)+f(3/2);

> plot({f(x),tf2(x)},x=0..2);

- Compute the derivatives of the following functions using both
the
`diff`command and`D`operator for each function. - Find the tangent line to the graph of
at and plot the function and the tangent line on
the same graph.
- We say that two curves are perpendicular at a point where they
intersect if their tangent lines at the point of intersection are
perpendicular. Show that the two curves and
are perpendicular at their two points of intersection. Include a plot
showing the two curves and the two tangent lines. Recall that two
straight lines are perpendicular at a point of intersection if the
slope of one line is the negative reciprocal of the other slope. That
is, if the slopes are and then the lines are
perpendicular at their point of intersection if .
- Consider the function

Find all of the points where the derivative of is equal to .

2001-01-30