- Algebraically as the following limit

- Geometrically as the slope of the tangent line to the graph of
at .
- Functionally as the instantaneous rate of change of at .

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 .

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

The following limit determines .

> 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.

Note the `diff` command is also available via the
context-sensitive menu for an expression. Just select the
`Differentiate` item from the menu. However, the menu route
will not work directly on functions and can not be used to obtain the
`D` command.

However, to save you some trouble, the

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 . 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* , but .

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);

- 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. - Given the function
,
- Plot between .
- Find the slope of the line tangent to the graph at .
- Find any other points on the graph of that have the same slope as the tangent line at .

- For the function given below,

- Find the equation of the line tangent to the graph of at the point .
- Plot the function and the tangent line on the same graph.
- Find a point on the graph for which the tangent line is perpendicular to the tangent line at . You need only consider .

- In the case that is the position of an object as a
function of time, the difference quotient

can be interpreted as the average velocity of the object during the time interval and the limit of the difference quotient is the instantaneous velocity. Suppose that the position in meters of an object is given by where is time in seconds.- What is the average velocity of the object over the interval ? Include a plot of and an appropriate secant line whose slope is equal to this average velocity.
- What is the average velocity over the time interval ?
- Is there a time between and at which the instantaneous velocity is equal to the average velocity over the time interval ?
- At what time is the instantaneous velocity equal to ?
- At what time is the instantaneous velocity equal to zero?

2000-09-26