** Next:** About this document ...
**Up:** No Title
** Previous:** No Title

- Algebraically as the following limit
- Geometrically as the slope of the tangent line to the graph of
*f*(*x*) at*x*=*a*. - Functionally as the instantaneous rate of change of
*f*at*x*=*a*.

You can use the definition and the Maple `limit` command to
compute derivatives from the definition, as shown below.

> f := x -> x^2+3*x+5;

> (f(1+h)-f(1))/h;

The following limit determines *f*'(1).

> limit ((f(1+h)-f(1))/h,h=0);

The following limit determines *f*'(*x*).

> limit((f(x+h)-f(x))/h,h=0);

Maple also knows how to compute the
derivatives of most functions. The main command for differentiating
functions is `D`, also shown in the examples below. The last
example shows how to use the `D` command to define a function
`df` that is the derivative of `f`.

> D(f);

> D(f)(1);

> df := D(f);

> df(x);

There is also a `diff` command for differentiating
expresssions. Some examples are given below.

> diff(cos(x),x);

> p := x^3+sin(x);

> diff(p,x);

> f(x);

> diff(f(x),x);

> subs(x=1,diff(f(x),x));

To learn more about how to use the `D` and `diff` commands,
see the help pages. In general, the `D` command is useful for
computing the derivative of a function at a point because it produces
a function. The output of the `diff` command, on the other
hand, is an expression. Expressions are easy to plot, but putting in
numbers to evaluate an expression requires the `subs` command,
as shown in the last command in the examples above.

The secant line with base point *x*=*a* and increment *b*-*a* of a function
*f*(*x*) is the straight line passing through the two points (*a*,*f*(*a*))
and (*b*,*f*(*b*)). Given this information, it isn't too hard to write
down the equation for the secant line, given a function, base point,
and increment as

The `secantline` command takes three arguments. The first is a
function or expression, the next one is the base point, and the third
is the increment *b*-*a*. Try the commands in the examples below to learn how
to use this command. If you want to learn more, consult the help
page.Note especially that the third argument to
`secantline` is *not* *b*, but *b*-*a*.

Also new in the examples is the Maple `
animate` command, which is part of the Maple `plots` package. You
must issue the `with(plots);` command before you can use animate.
The `animate` command first produces what looks like an ordinary
plot in the worksheet. However, if you click on the plot to make it
active, controls for the animation appear in the `context bar`
that are similar to
those on a VCR. You should be able to figure out how they work by
experimenting. In the examples below, the animate command shows the
tangent line as the limit of secant lines.

> with(CalcP);

> f := x -> x^3-2*x+1 ;

> secantline(f,x=0,1);

> secantline(f,x=0,0.5);

> plot({f(x),secantline(f,x=0,1),secantline(f,x=0,0.5)},x=0..1);

> with(plots):

> animate({f(x),secantline(f,x=0,1.5-t)},x=-0.5..1.5,t=0..1.49);

> secantline(f,x=0,h);

> limit(secantline(f,x=0,h),h=0);

- 1.
- Find the derivatives of the following functions three different
ways using Maple. From the definition, using the
`D`command, and using the`diff`command. Note that Maple has its own ideas of what form the answer should take, so some of your answers may look unfamiliar or be more complicated than you think they should.- (a)
- (b)
- (c)

- 2.
- Given the function
*f*(*x*) =*x*-0.5^{4}*x*-6^{3}*x*+ 0.5^{2}*x*+5,- (a)
- Plot
*f*(*x*) between . - (b)
- Find the slope of the line tangent to the graph at
*x*=0.5. - (c)
- Find any other points on the graph of
*f*(*x*) that have the same slope as the tangent line at*x*=0.5.

- 3.
- For the function given below,
- (a)
- Use
`Maple`to define the function as a function of x. - (b)
- Use
`Maple`to define the difference quotient as a function of h. - (c)
- Evaluate the limit of the difference quotient and compare
this result with the derivative of the function using the
`Maple``D`command. Explain why you are grateful to have`Maple`to answer this question. - (d)
- Find the equation of the line tangent to the graph of
*f*(*x*) at the point . - (e)
- Plot the function and the tangent line on the same graph.

- 4.
- Suppose that
*s*(*t*), defined as for , describes motion of a particle in one dimension. That is,*s*is the distance of the particle from some reference point as a function of time.- (a)
- Find the secant line of the graph of
*s*between*t*=1 and*t*=2. - (b)
- Describe how you can use your answer to the first part of this exercise to compute the average velocity of the particle over the time interval . Report the value of the average velocity.

9/21/1999