Motivating Questions for Calculus * Martin Pergler, Oct 1998

Sure, calculus deals with concepts called "derivatives", "integrals", "limits", even "differential equations", and we'll learn what all this means and how to compute with it. But it's natural to ask "why should I care?" and "what are these concepts really good for?"

Section 1.1 of the textbook gives a good overview of some of the *ideas* behind these concepts, and maybe that discussion already makes you feel "this is interesting. I want to know more about it."---a form of intrinsic motivation. Here I try to give some extrinsic motivation---via questions we can state without using calculus, but where calculus (or related math we will discuss) is very useful for answering them. I call them "motivating questions for calculus". Perhaps some will strike your interest. I'm not saying you should be able to answer them all (and if you can answer many of them already, then you're in the wrong course!) or ever understand right now what all of them are asking, but we will return to them all over the course of the year to see how calculus applies.

Some of the questions are explicit word problems, and are meant as representative of a class of problems you could solve by similar techniques. Others are more vague philosophical questions.

1. (Optimization) An island lies 30 miles off a straight coastline. A cable is to be laid to connect the island to a switching station which is on the coastline, 100 miles away from the point closest to the island. Underwater cable costs $5,000/mile and land cable costs $3,000/mile. How should the cable be laid to minimize cost?
2. (Speed) Two Olympic sprinters run 100m in exactly 10 seconds. One is ahead at the beginning, but the other catches up in the end. I claim that each is running at an instantaneous speed of 10 m/s at some moment. Why? What does instantaneous speed mean anyway? I also claim that at some instant, both running at the same instantaenous speed (but not necessarily 10 m/s). Why?
3. (Modeling motion) You drop a rock out the window. What does the graph of its height off the ground versus time look like? Why? What if you drop a feather, or other object with large air resistance? How can we model this mathematically?
4. (Population biology, etc.) A population of animals (say rabbits) with no predators and unlimited resources (food, space...) increases at a rate which is proportional at any moment to the size of the population at that instant (# baby rabbits is proportional to # mother rabbits). How can we model this using a mathematical formula? What if the resources are limited, and so that the birth rate decreases as the population gets large?
   [The same math is used for modelling chemical reaction rates and intravenous drug dosing.]
5. (Strange economic phenomena) You invent a new gadget and manufacture it. As you continue making it, you gain experience and are able to make more and more cheaply. Quantify this mathematically. [This is called the experience curve, and the form it often takes is a thorn in the side of some impressive economic theories.]
6. (Sound and music) How do people identify the pitch of a musical note, and what instrument is playing?
7. (Drawing curves) Many computer graphics packages have a command to draw a smooth curve connecting several specified points (often called "spline", "freehand curve" or something similar). How does this work?
8. (Weird mathematical facts)
   [A] The number [square root of 2] is irrational (i.e., cannot be written as a ratio of integers). Why?
   [B] What do we mean by 2^sqrt(2) [2 to the power square-root-of-2], or 2^x for other irrational numbers x?
   [C] What sort of number is [pi] (the circumference of a circle / diameter) ? How can we calculate it?
9. (Sequences and series) Make sense of the following claims:
   [A] 0.99999999....=1 [B] 1+1/2+1/4+1/8+1/16+....=2 [C] 1+1/2+1/3+1/4+1/5=infinity
   [D] 1-1/2+1/3-1/4+1/5-+...=[Area in the x-y plane between x=1, x=2, y=0, and y=1/x] = 0.693147...
10. (Zeno's paradox) The speedy Achilles challenges a tortoise to a race, and graciously gives it a head start. He only starts running when the tortoise has plodded to a tree 100' ahead. But by the time Achilles reaches the tree, the tortoise has plodded another couple of feet farther. When Achilles reaches that point, the tortoise has managed another few inches, and so on ad infinitum. So the tortoise is always ahead of Achilles. What's wrong here?
11. (Forbidden but allowed) People say "you can't divide by zero". Why can we sometimes do it anyway?
12. (What is math research) Mathematicians spend their time "proving theorems". What does this mean, and how has it changed during history? What is it good for (other than giving mathematicians something to do)? How are computers useful for "doing higher math"?
13. (Awe of Calculus) "Calculus is one of the fundamental achievements of Western Philosophical Thought." Truth, bunk, or hyperbole? How did this achievement (whether fundamental or not) take place?