Calculus Lab: Zooming In

Goals

This lab helps motivate the concepts of limits and the slope of a function. Specifically, the lab provides experience with:

• exploring the shape of a function locally by progressively magnifying the function near a point,
• defining the slope of a function at a point by zooming in on that point, and
• identifying examples, in which the slope of a function is not defined.

Acknowledgement

This lab draws heavily upon a lab by a similar name in Learning by Discovery: A Lab Manual for Calculus, edited by Anita Solow, MAA Notes, 1993. In particular, much of the approach here comes from a 1993 lab, although all references to Matlab and corresponding procedures are new.

This lab uses the graph2d interface, written by Professor Emily Moore for the Matlab mathematical software package.

Background

Within analytic geometry, the slope of a non-vertical line is often computed as follows:

• Pick two distinct (x₁, y₁) and (x₂, y₂)
• Compute the quotient (y₂ − y₁) / (x₂ − x₁), the change in y values over the change in x values. This quotient is sometimes called the rate of change of line.

Calculus asks the question of how we might extend the concept of rate of change to functions that are not straight lines.

In this lab, we will experience a key observation: when we zoom in on many functions, the functions look very much like straight lines. This observation will allow us to estimate the slope of a function at many points.

Getting Started with MathLAN and Matlab

For various reasons related to specific software applications and technical capabilities, both the Department of Mathematics and Statistics and the Department of Computer Science at Grinnell College use Linux-based computers. These machines run on a Local Area Network (LAN) called, MathLAN. Although MathLAN provides extensive processing opportunities, we will be able to avoid many technical details throughout this course. Feel free to talk to me if you would like to know more about MathLAN and its capabilities!

MathLAN computers are maintained separately from the college-wide machines maintained by Information Technology Services. In particular, although you normally will have the same username for both systems, passwords are maintained separately.

Within MathLAN, we will use Matlab, a powerful software package for numerical processing.

To get started, follow these steps:

1. Log into a MathLAN workstation using your MathLAN password. (These passwords will be distributed during the first lab session.)

2. If you wish, you may change your password to anything of your choosing, as follows:

   1. Open a terminal window, by clicking on the icon of a computer monitor in the menu panel at the bottom of the screen.
   2. Move your mouse to the newly-opened terminal window, click in that window with the left mouse button, and type

      
      password
      

      You will be asked for your old password and then offered the opportunity to type your new password. (You are asked to confirm the new password to help reduce the risk of typographical errors.)

3. Start Matlab by clicking on the icon with 0's and 1's in the menu panel at the bottom of the screen. A new window will appear, providing an initial text-based interface for Matlab.

4. For this lab, we will use the graph2d option; type

   
   graph2d
   

   at the >> prompt, and follow the instructions in the graphics window that is created.

Exploring the Function f(x) = 2x² + 8/x - 7 on the Interval [1, 3]

As an initial experiment, we will examine the function f(x) = 2x² + 8/x - 7 near the point (2, 5).

5. Within the Matlab graphics window, graph f(x) as follows:

   1. Enter the function in the f1 box as

      
      2 * x ^ 2 + 8 / x - 7
      

      (Be sure to use a lower-case letter x, not upper case.)

   2. To get a general view of the shape of this graph, set xmin to -2 and xmax to 8.

   3. Adjust the vertical scale by experimenting with values for ymin and ymax.

   To check your entry of the function is plausible, check that the graph contains the point (2, 5).

   Write a paragraph (at least four sentences) describing the overall shape of this graph.

The graph2d utility within Matlab provides two capabilities that are particularly useful for our experiments:

• You can find the coordinates of a point on a graph by clicking on the point. The point's coordinates will appear in the box on the upper right of the graphics window.
• You can zoom in a point by clicking on the zoom box and then clicking on your target point within the function.

6. For this function y = f(x) = 2x² + 8/x - 7, zoom in on the point (2, 5) until the graph looks reasonably straight. Then pick a point on the curve other than the point (2, 5), and estimate the coordinates of this point. Calculate the slope of the line through these two points. This slope is also called the derivative of f at x = 2, and is denoted f'(2).

   • After clicking on the "zoom" box, you can zoom in multiple times by using the left button — clicking on the point you want to target.
   • When you have zoomed in as far as you wish, click the "zoom" box again. Then, clicking on a point will display the coordinates of that point. This will help you identify points for computing an approximate slope.

7. With your answer to step 6, you know the slope of a line through the point (2, 5).

   1. Compute the equation of this line in the familiar slope-intercept form: y = mx + b. (Check the equation of your line to be sure that it really contains the point (2, 5).)
   2. Enter this line into graph2d as a second function f2, and graph the result.

   (If the line does not seem reasonably close to the original function y = f(x) = 2x² + 8/x - 7, check your computations and/or talk to the instructor.)

8. Zoom out a bit from (2, 5), and write a paragraph describing any relationship you see between the original function and this line.

Some Additional Functions and Lines

9. For each of the following functions, graph y = f(x), on the indicated interval, zoom in on the specified point, estimate the slope of the function at this point, determine the equation of a line through this point with the given slope (in slope-intercept form), plot the line as well as the function, and describe what you see in a few sentences.

   1. f(x) = x⁴ - 3x² on the interval [-2, 2] and the point (1, -2).

   2. f(x) = cos x on the interval [-π, π] and the point (&pi/2, 0).
      (Note, in Matlab, enter cos x with parentheses as cos(x). Also, the value π/2 may be specified as Pi/2.)

   3. f(x) = cos x on the interval [-&pi/2, π] and the point (0, 1).

   4. f(x) = (x - 1) ^1/3 on the interval [-1, 3] and the point (2, 1).
      (Note, in Matlab, the cube root function, x^1/3 should be written as cbrt(x), so that graph will appear correctly.)

More Experimentation

10. Graph f(x) = (x - 1)^1/3 again. This time, zoom in on (1, 0). Describe what you see. By examining your graphs, explain why the slope is not a finite real number at x = 1.

11. Graph the function f(x) = | x⁴ - 3x² | on the interval-2, 2] By looking at the graph and zooming in on points you select, decide at which points the function f has a slope or derivative and at which points it does not. Explain why f does not have a slope or derivative at some point you selected, and draw a sketch that supports your explanation.
    (Note that the absolute value function, | x |, is written abs(x) in Matlab.

Work to Turn In

• Commentary for steps 5, 7, 9, 10, 11 (with sketch).
• Computations for steps 6, 8, 9.

This document is available on the World Wide Web as

http://www.cs.grinnell.edu/~walker/courses/131.fa09/labs/lab-zooming.shtml

created 5 June 2009 last revised 30 August 2009
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.