Here are some of the many activities that I have written.
They are intended for use in a calculus class, and their
topics range from an introduction to functions
from numerical, graphical and analytical perspectives (``A brush
with death!!'') to solids of revolution (``Cooking with mathematics''),
work (``The drink of the gods'') and vector fields
(``Vector Field Battleship!!'').
You have been sent to the exotic city of Uxmal to
investigate the annihilation of its entire population.
It is believed that a plague infested the region and wiped out all
human life. Yes, it is a most dangerous mission,
but that didn't stop you from leaping at the first
opportunity you found to get away from the
monotonous life that you'd been leading lately.
You've been provided with all the necessary information and equipment for this assignment, but even that is no guarantee of success. The reason why they put you on this mission and not one of their top micro-biologists is that you had done something very similar in a calculus class when you were in college, and that had turned out great.
Once in Uxmal, your first stop is the micro-biology lab at the university; if there is any useful information at all, it must be there. The people who used to work at the lab had managed to gather a great deal of information, though not enough to stay alive. They knew that what was behind all the deaths was indeed a lethal virus. Several hours after being exposed to the virus, a rash appeared on the patient's skin, and at that precise time the virus started to produce a toxin. The toxin remained in the body for only 24 hours, but if (when) it reached a level of 3 points, the patient died. They had also developed a partial antidote, but even the strongest dose was not enough to neutralize the toxin, and in the end they all died in pain.
After reading such depressing news you just wanted to call it a day, and the only thing that stopped you was the rash that started to sprout on the backs of your hands, spreading quickly to the rest of your skin... You have been infected in spite of your anti-viral suit and all the fancy equipment that was supposed to protect you!! Now you have probably the best possible motivation to solve this problem.
Here are the facts that they had obtained (the numbers presented here have been rounded to the nearest 0.01, and are only approximations):
The toxin level x hours after
the rash appears, denoted
by f(x), had the following behaviour:
The heaviest dose of antidote that can be safely
administered (and this can only be done once for
each patient) can neutralize only part of the toxin,
denoted by g(x) (where x is now the number of
hours after the drug was injected) and illustrated by the table:
The DNA structure of the virus shows that it belongs to a
special subfamily of the Third Order Homogeneous Linear type,
which means that the function f satisfies the differential
equation f'''(x)=0, which (as we shall see later in this
course) implies that f(x) is a quadratic polynomial. You also
know that f(x) is only defined for the interval [0,24], that
there's no toxin at the beginning or the end of the 24 hour period,
and that f(1) is exactly 23/15.
1. What is a formula for f(x)?
The molecular structure of the antidote is very similar to
that of the virus, and an idea slowly forms in your mind.
2. What could this mean in terms of g(x)?
You also know that the antidote works for only 20 hours,
that its effect is gradual,
and that one hour after it's injected the amount of toxin that
is being neutralized
is exactly 19/15.
3. What could an explicit formula for g(x) be?
4. Is there any way I can check this?
Now that you know what functions you are dealing with, you want to
know more about them. The fastest and easiest way to get a lot
of information about a function is to look at its graph.
5. What do the graphs of f and g look like?
Once you looked at both graphs simultaneously you suddenly hit
on a solution to the problem (remember, the key to surviving is
to keep the toxin level below 3 points for the 24 hours following the
appearance of the rash).
6. How can you save your life and perhaps countless others?
You've done it!! Now all you have to do is finish your report. Here's the information your superiors will need:
7. What were the toxin levels in your body (hour by hour) after you broke out in a rash?
8. What exactly were the toxin levels at each moment for the 24 hours following the eruption of the rash?
9. How would you describe the function whose graph you just gave?
10. Give estimates of the earliest and latest times when the antidote can be given to save a patient.
11. Is it possible to get exact values for the previous question? 12. How?
13. The activity that you've just finished shows one of the many
ways in which you can use functions to solve problems and
have some fun. Now we'd like to invite you to have fun
creating your own adventure. Among the endless possibilities
that you may want to consider as sources
for functions and plots are: radioactive materials,
space travel, weapons, war, love, food,
poison, money, air pressure, light, heat, wind, et cetera.
You may want to add a twist (or two) to your story by choosing
functions with special properties.
Whatever you do, remember that there is only one rule: enjoy it! If you're working in a group, then some of the tasks can be done by different members, so you can come up with a more interesting piece of literature (this is the relatively unknown genre of ``fictional science'', which can only be fully appreciated by those who master its theoretical subtleties). Once you're done, swap stories with another team, and see who can solve the other's puzzle first.
I'd also like to take this opportunity to ask for your input on this activity: did you like it?, was it useful in learning the material?, would you like to do something similar again?, do you have any other suggestions? Remember that you are the most important part of this learning experience, and your opinion is very important to all of us.
We'll use solids of revolution to model food.
1. Decide (as a group) which of the following items can be described using solids of revolution: a doughnut, a cup of cocoa, a turkey, a square cake; a pretzel, a bagel, a watermelon, a sausage, a curly fry.
Now we're going to make bagels, but instead of kneading dough for countless hours, we'll simply use solids of revolution. As every cook knows, you need to rotate a circle in order to make a bagel, but the subtleties of rotating circles often elude novice cooks. Here's the most common mistake when making bagels. Describe what the problem is and how to correct it:
2. Circle of radius 1 centred at (1,0) rotated around the y-axis.
The ``standard'' bagel has always been the one obtained by rotating
the circle of radius 1 centred at (1,0) around the line x=-1.
Recently, a French chef created a bagel with the same
circle but rotated around the line x=-2.
3. Physically, what's the difference between the two bagels?
4. What are the volumes of the standard and the French bagels?
5. Create your own bagel by rotating the circle of radius 1 centered at (1,0) around the vertical line of your preference. Find the volume of your creation.
6. Swap answers with another team. The first one to uncover the other team's recipe wins (this may be much quicker than you think!)
7. What is the ``winning strategy'' in the previous game?
The year is 2004. You're working for a very successful company
that specializes in fine foods and drinks. Their latest
creation, ``nectar'', sells at $100 a bottle (which contains
only 0.05 pounds of nectar!). Today is just another ordinary day: a robot
had just taken a tank filled with nectar to the bottling area, and
you are about to recharge its battery like you've done so many
times before. Then it happened. It came without a warning, and
it was over before anyone could do anything about it. The lid on
one of the tanks was blown up (perhaps sabotaged by
an unscrupulous rival firm), and due to the extremely high
volatility of nectar, this could have dire consequences, unless
someone like you comes along to save the day.
The tank weighs 1000 pounds, and it has to be moved 100 feet to the bottling area, which takes a work of *** foot-pounds assuming that the weight remains constant. In reality, pure nectar evaporates at a constant rate of 10 pounds per minute, and the tank is moved at a constant speed of 10 feet per minute. This means that if the tank initially weighs 1000 pounds, after moving it 10 feet it'll weigh *** pounds, after x feet it'll weigh *** pounds, by the time it reaches the bottling area it'll weigh *** pounds, and the work required to move it was *** foot-pounds.
As if the loss of *** pounds of nectar were not bad enough, the robot is not fully recharged, and it can only do 80,000 foot-pounds of work. Since there are no other ways of moving the tank, you'd better use the robot wisely, or else you'll lose all the nectar. The manager decides that the best thing to do is wait until the tank weighs 800 pounds and then use the robot, so the work will be less than 80,000 foot-pounds. In all fairness, this is a good conservative plan, but you have a much better idea.
What is your idea?
If the company gave you 50% of what you saved them, how much money would you get?
Rules: The game is played between two teams, and each team will
be provided with a secret two-dimensional vector field.
Flip a coin to decide which team plays first. The first team
asks their opponents the value of their secret vector field at
a point inside the square [-10,10]x[-10,10], and then
will proceed to draw a vector on their map indicating that
piece of information. Then it's the other team's turn to ask
the value of the vector field at a point, and the teams will
continue to take turns until either one of the teams hits a
point where the vector field is (0,0) (and instantly wins the game)
or time's up. If the game ends before either team finds the
point where the other team's vector field vanishes
(that is, where it's (0,0)), then each
team will guess where that point is, and the team that
gets closer (in norm) to the actual answer wins.
Remarks: During the first phase of the game, both teams will have one stable critical point and the arrows will be pointing toward the critical point; during the second phase (if there's enough time for a second phase), both teams will have one unstable critical point, and the arrows will be pointing away from the critical point.
Have some mathematical fun!!