Calculus Tutor

Here are some cool experiments to do yourself or to use when explaining calculus to students.:

 The limit as illustrated with a fan.

 This experiment is useful during the discussion of evaluating limits. Students often wonder why one can't always evaluate a limit by just plugging in a value.

An electric fan is set up with no safety shield. A pinwheel serves as a wind meter to measure windspeed (in miles/hour) f(x) at a distance of x feet from the fan. The following readings are taken:

f(2) = 15
f(1) = 17
f(.5) = 18.3
f(.2) = 19.2
f(.1) = 19.6
f(.01) = 19.8
f(.001) = 19.98

so it appears that

  lim (x->0) f(x) = 20

Demonstrate what happens when you try plugging in the value x=0. This should be done so as to send the pinwheel flying across the room without allowing a reading to be taken, showing that you cannot simply plug in the value to find a limit. If you have some ketchup packets in your hand, which you squeeze as the pinwheel flies, you can make an even better demosntration of the danger of plugging in.

 Bring a collection of cylindrical cans and containers to class. Bring in a variety of cans such as coke cans, soup cans, tuna fish cans etc. Ask the class to guess which ones are the most efficient. Generally the optimal shape (a square cross section) is found on Borden evaporated milk cans and little else. Calculate what percentage of the material going into some of the other cans is wasted.

You could also bring in detergent boxes and other rectangular shaped containers and do similar experiments. Boxes holding oranges, detergent etc can be fairly arbitrary in dimension so they make good examples. Cardboard cartons with no tops which are folded together so that certain sides are made of double thick layers make nice examples where the answer is not obvious, and can be compared Jo an actual box.  

 Problems on speed and acceleration can be illustrated by bring balls to class, throwing them in the air, and using them for generating the data e.g. throw a ball so that it reaches the top of the board, five feet above where you release it at the bottom of the board. Ask the class to calculate the initial velocity. For a variation, use an egg instead of a ball. Make sure it is hard boiled, or else use a toy egg which will break into two pieces when it hits the ground. You might break a real egg into a cup at some point to add some theater.  

Construct a curve using a wire hanger or some other type of wire. Slide a straight stick or rod along the curve using two washers. As the two washers come close together the slope of the line approaches the derivative. 

 Three dimensional bodies such as spheres, ellipsoids and surfaces of revolution can be demonstrated by examples. Fruits such as carrots, tomatoes, watermelons can be used for this. Cutting them up with a sharp knife illustrates sections of these surfaces. More dramatic demonstrations can be made with an axe. Most examples can be passed around on plates following the demonstration. While this may involve significant amounts of expermimental material for a large class, students seem to exhibit great appreciation for examples that can be consumed.

A warped tennis racket nicely illustrates the graph of y²-x². To make one, get an old wooden racket and leave it outdoors in the rain for a few weeks. Curled up leaves also form suitable examples.

Even elementary facts such as that two planes intersect in a line are more apparent to a substantial number of students if illustrated by simply made models. Glue together two pieces of stiff cardboard to show two planes intersection.

Glue a pencil perpendicularly to a plane to show how a plane is determined by a point and a normal vector. 

 Variation 1: You need a coffee pot for this to get some water boiling hot. Add some hot cocoa mix, or tea. State that experiment has shown that hot cocoa at above 135 degrees burns your tongue. Use a thermometer to measure the temperature - say you get 170. A meat thermometer works well wtih this. Alway prepare the calculations ahead of time, as you won't have much time to do this in class. After a minute passes, remeasure it and get, say, 155. Then calculate how long it will take to reach 115. When the time arrives, gulp down a lot of the hot liquid to show your faith in your calculation. If you have miscalculated, hold your composure as you do not want to shake your students' confidence in calculus.

Variation 2: Bring in or boil up a hot cup of black coffee in one cup and a small amount of cold milk in another. Ask - "If I will be serving the coffee in five minutes and want it to be as hot as possible, should I mix the milk and coffee together now or just before I drink them?". Try it both ways and see.  

 For a realistic related rates problem, get a radar gun which measures the speed at which an object is moving towards the gun. Have students throw a baseball or tennis ball at a 45 degree angle from the gun's line of sight. Calculate the ball's speed using related rates. (Getting hold of a radar gun may be difficult.)

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