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Re: [Phys-l] Work and Energy: which first?



One way to determine the area under a non-linear function, without calculus, is to use a uniform graph paper and a sensitive electronic scale. Suppose the curve is drown on a graph paper using some units, such as newtons versus meters. Suppose the scales chosen were such that 1 cm^2 corresponds to 1600 J. Using a pair of scissors the area under the curve is cut. Also cut is the area of a 4 by 4 cm square. Suppose the weight of the square is 320 mg while the weight of the area under the function is 160 mg. Then we know that the area under the curve corresponds to 12800 J.

This is also a good exercise on "proportional reasoning." The suggested method would produce a wrong result if the surface density of the paper were not uniform. That is also worth emphasizing, for example, by asking "under what conditions can the result be wrong?" With a good scale the "weight method" is probably going to be more reliable than the "area method" in which tiny squares (or triangles) are counted below the curve. Both methods belong to the non-caluculus category.

Ludwik
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On Jun 13, 2006, at 1:58 PM, Hugh Haskell wrote:

At 13:17 -0400 6/13/06, Robert Cohen wrote:

I am also intrigued by your statement that the area-under-a-curve method
"[brings] in calculus ideas without the formalism." Doesn't the
average-force method also bring in the same calculus ideas?

I am highly in favor of introducing calculus ideas without the
formalism. In fact, I never mention the word "calculus," or related
terms, when I am showing the students how to calculate the area under
a non-linear curve, or how to use a spreadsheet to, in effect, solve
a differential equation. But they get to see a lot of calculus
without realizing it, and since it isn't called "calculus," they
don't get freaked out by the fact that they are supposed to be in a
"non-calculus" course. Later, when they study calculus, and see the
ideas they saw before introduced in an explicitly calculus context,
they are ready to accept the development without the panic that would
ensue if they realized that I had been "sneaking" calculus in earlier.

But I would argue that using the average doesn't bring in calculus
ideas to the extent that using the area under the curve does,
especially for a non-linear curve. Using the area to find the average
for a non-linear curve enables the students to see that, for a rising
curve the later values have more weight than earlier ones, and for a
descending curve the converse is true. For an arbitrary curve, they
will see that a large value of the ordinate will count more that a
small value, over the same range of abscissa.

Once they understand that, it isn't much of a jump to the realization
that when you draw a horizontal line at the value of the average, the
area between the average line and the part of the actual curve
*below* the average must equal the area between the average line and
the part of the curve *above* the average, if the average is chosen
correctly. That is easily seen with a straight line graph, so that is
a good place to start.

The point is that both ways of visualizing the average should be taught.

Ludwik Kowalski
Let the perfect not be the enemy of the good.