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Next: The linear approximation Up: Labs and Projects for Previous: Limits of functions.

Definition of the derivative

Purpose

The purpose of this lab is to use Maple to explore the geometric and algebraic aspects of the derivative.

Background

The derivative of a function at a point x=a, often written , can be interpreted in several different ways.

Probably the second and third interpretations are the most important; they are certainly closer to what makes the derivative useful. In this lab, we will use Maple to explore each of these different aspects of the derivative.

You can use the definition and the Maple limit command to compute derivatives directly, as shown below. Maple also knows 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.

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

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

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

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

  > 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);

To learn more about how to use the D and diff commands, see the help pages.

The secant line with base point x=a and increment b of a function is the straight line passing through the two points and . Given this information, it isn't too hard to write down the equation for the secant line, given a function, base point, and increment. However, to save you some trouble, the secantline function has been written. This function is not a standard part of Maple, but is one of about thirty Maple commands that have been written at WPI for calculus. In the examples below, the first command shows you how to load the CalcP package containing these commands. You must do this before you can use secantline. If the output from the with(CalcP); command is different from what you see below, or the secantline command doesn't seem to work, ask for help.

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.

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);

Exercises

  1. Describe how the commands in the last set of Maple examples above illlustrate the definition of deriviative as the limit of the slope of a secant line.
  2. Find the derivatives of the following functions using Maple, both from the definition and using the D command.
  3. Determine the derivative of the function defined by

    for all values of x for which exists. Describe how you obtained your results.



next up previous
Next: The linear approximation Up: Labs and Projects for Previous: Limits of functions.



William W. Farr
Mon Jun 26 13:37:30 EDT 1995