Calculus Lab: 4/12/2022

Today's Lab consists of two parts: 
(I)  A set of exercises from the textbook that involve applying derivative concepts in a variety of settings.  
(II)  The coolest graph contest!

For this lab please turn in a homework-style report that primarily focuses on the work you did for each problem. However, please be mindful that I do expect complete and clearly organized solutions, integrated with discussion or interpretation of your results where needed.

(I) Textbook exercises

 Work out complete solutions to these exercises on pages 237-240:

        5, 6, 11b, 13c, 16c, 24, 28, 33, 34, 35.

and these exercises on pages 245-246:

        17, 25, 30, 36.

(II) The coolest graph contest

 (This is NOT an optional part of the lab -- it will count for 10% of the grade.)

We are sure you've seen more than your fair share of cool graphs!
And we know that in the vast realm of all things cool, graphs wouldn't exactly top the list. But still... we hope this contest is not only insightful, but also entertaining and gives you a new appreciation for equations and graphs!

Instructions: Use the Apple Grapher or Sage (see the sample scripts below) or Desmos to explore the following examples, and to try out your own variations in an effort to make your own coolest graphs. Turn in any two of your own variations to enter into the coolest graph contest! On your printout, or in your lab report, give the specific equations you used to create your graphs.

  1. \(x = 0.02 t \cos(t)\),   \(y = 0.02 t \sin(t)\). Use \(t=0\) to 10, and \(t=0\) to 100.
    Variations: Replace one of the 0.02 by 0.04.
                    Replace \(\sin(t)\) by \(\sin(2t)\) or \(\cos(t)\) by \(\cos(2t)\).
                    Interchange and/or replace \(\sin\) with \(\cos\).
  2. \(x = 2 \sqrt{\cos(20t)} \cos t\),   \(y = 2 \sqrt{\cos(20t)} \sin t\). Let \(t=0\) to 100.
    Variations: Replace cos(20t) with [cos(20t)]^(1/20)
  3. \(x = t + \sin (2t)\),   \(y = t + \sin (3t)\)
    Variations: Replace sin with cos. Play with different numbers.
  4. \(x = 4 \cos t\),   \(y = 2 \sin (t + \sin (150t))\)
  5. \(x = 1.5 \cos t - \cos (30t)\),   \(y = 1.5 \sin t - \sin (30t)\)
  6. \(x = 4 \sin (t + \cos (100t))\),   \(y = 4 \cos (t + \sin (100t))\)

  7. y2 (y2 - 6) = x2 (x2 - 8)
  8. y2 = x3 + 3 x2  

 
The following Sage script shows an example of how to plot an implicit curve. Note that it is way easier to do the same plot using the Apple Grapher or Desmos.

 
This next Sage script shows how to plot parametric curves.

 
Check out the following graph I made while fooling around one idle day!

Also check out these waaaaaaaa... cool graphs (look at Sec. 2) made by some of our competitors just down the road at Kenyon College!