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- 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 pops up a separate window with controls like
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=1,1-t),x=0.5..2.5],t=0..0.99);

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

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

- 1.
- Given the function
*f*(*x*) = 0.01(7.5*x*-7.3^{4}*x*-8)-3^{3}*x*,- (a)
- Plot
*f*(*x*) between . - (b)
- Find the slope of the line tangent to the graph at
*x*=2. - (c)
- Find the slope of the line normal to the tangent at
*x*=2. - (d)
- Find the point(s) on the graph of
*f*(*x*) that has the same slope as the normal line. - (e)
- Find the equation of the line tangent to the graph at the point found in part (d) and also the equation of the line tangent to the graph at
*x*=2. - (f)
- Plot the function on the same graph as the two tangent lines found in part (e).

- 2.
- Find a formula for the equation of the line tangent to the graph of a function
*f*(*x*) at a given point (*x*,_{1}*f*(*x*)) using the point slope-form of a line and the limit of the difference quotient for the slope. For each of the functions given below:_{1}- (a)
- at
- (b)
- at
*x*= 3.

- i. Use
`Maple`to define the function as a function of x. - ii. Use
`Maple`to define the difference quotient as a function of h. - iii. 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. - iv. Use the formula derived above to find the equation of the line tangent to the graph of the function at the indicated point.
- v. Plot the function and the tangent line on the same graph.

- 3.
- Given that the position of an object moving along a straight line at any time
*t*is given by the function .- (a)
- Use the maple
`plot`command to graph the function and find exactly how many times the graph changes direction and use the Maple`solve`command followed by the command`allvalues(``)`to find the exact number of roots. Find the velocity function and all t values such that the velocity will be zero. Plot the velocity function and using this graph, find intervals where the object is advancing (moving forward) and where the object is retreating (moving backward) and where the object has stopped for an instant to change directions. Explain these results in relationship to the roots of the position function and the roots of the velocity function.

9/22/1998