A friend suggested I look at "Student Assessment in Calculus", a Report of the NSF Working Group on Assessment in Calculus, published by the MAA and written by a group chaired by Alan Schoenfeld. The usual messy example of a calculation is given as typical of a traditional calculus question. Here it is to calculate the derivative of x^2 * exp(cos(x^3+5x+3)). Example 4 is the following. 1. Bring up "Derive" and use it to graph the function sin 2t cos t on -pi,pi. Guess the value of the integral of this function on that interval. Use "Derive" to calculate this integral, and two similar ones, sin t sin 2t and sin t cos 2t. On the basis of the three calculations above, make a conjecture about integrals of products of sine and cosine functions over this interval. Calculate two more integrals to check this conjecture. If this does not suport your conjecture, make a new one and test it with new examples. Continue until you are sure your conjecture is correct. 2. Use Derive to calcuate the following integrals: [all over -pi,pi] sin^2(t), cos^2(t), sin^2(2t), cos^2(2t). Make a conjecture, check it until you are sure it is true. Does your conjecture about integrals of squares (a kind of product) contradict your conjecture about products? Why or why not? 3. The student enters a number for the date of birth and a number for the month of birth, and a different computer system gives a function of the form a + b cos t + c sin t + d sin 2t, but hides it so that the student does not know a, b, c, or d. The student is to use what has just been done to find these coefficients, draw the graph and when it matches the one on the computer is done. A five page report is written, four of them dealing with the questions above, and the fifth will be sumarized below. What is wrong with this? Tom Romberg told me that when George Polya was teaching a class, he would regularly have students make guesses or conjectures, and state the evidence behind their answer. He would then ask the student if this was sufficient evidence. If the student said yes but had not given a proof, Polya would say NO, a proof needs to be found. I see no evidence that these students are even told that more is needed. The fifth page had the students, Doug and Dave were the names given them, writing out the values of some other integrals. These are the integrals of powers of sin t over -pi, pi divided by pi. Power Coefficient 2 1 4 3/4=.075 6 5/8=0.675 8 35/64=0.546875 10 63/128=0.4921875 12 231/512=0.451171875 14 429/1024=0.418945312 16 6432/10384=0.39276123 18 12155/32708=0.370941162
At the top of the page, the students had written: "As we mentioned above, the value of the area under the curve described by an even power of a sine or cosine function is pi times some fractional coefficient. In an independent exploration, I sought some way to calculate the coefficients of pi without actually evaluating the integral of the trigonometric function (over the interval from -pi to pi). I was unable to discern any definite pattern among the coefficients, but I was able to approximate the coefficients as a function of the power of the trigonometric function." After the numbers above, this continues with "The graph of coefficients versus the power of the trig function seemed to have some sort of exponential or power relationship. A log-log plot changed the graph to a straight line of the form y = mx + b, indicating the power relationship. Using MathCAD I solved for the slope of the line and its intercept and obtained an equation for approximating the coefficient of pi as a function of the power of the trig function.
coefficient = c^(0.373) * power -(0.474).
Note that this is only an approximation; however, the differences between the approximations and the actual values for the coefficients is small. More importantly, this approximation saves time in calculating the integral of an even-powered trigonometric function when the power is large."
Richard Sisley has argued that most polynomials can not be factored, which is true. He could have extended this argument a bit and said that most mathematics problems can not be solved exactly, so why not just use numerical methods, as the student did above. The problem with this approach is that very important tools will not be developed. Consider this integral, and the values given as rational numbers (except for the last two, which are incorrect). If you try to make sense out of these numbers, and not out of the decimal versions of them, it is natural to see how you go from one number to the next. These numbers are 1, 3/4, 5/8, 35/64, 63/128, which is going to be enough to see what the pattern is. While there is no obvious pattern to the numbers as given, it you divide one number by the previous number you get 3/4, 5/6, 7/8, 9/10. The pattern is now obvious, and can be proven by integration by parts. There are two ways to write the answer, the usual way it has been written in traditional calculus books, and the way it should be written to lead to the next step. The way it has traditionally been written is to multiply the factors obtained, and get
1*3*5*...*(2n-1) ---------------- 2*4*6*...*(2n)
A better way to write it is to use the following notation, which gives the analogue of x^n when considering finite differences.
(x)_n = x(x+1)...(x+n-1).
The formula above can be written as (1/2)_n ------- n! . Euler, and later Gauss, considered such ratios. This one goes to zero like a constant times n^(-1/2), and the constant is important. In the general case, (x)_n lim ----- n^(1-x) exits and occurs often enough to merit n->inf n! a name. It is called 1/gamma(x). The gamma function, which I will write as G(x), is the natural extension of the factorial function. In our classes, it is usually introduced via an integral, but there is a lot to be said for introducing it as this limit, which of course you have to show exists. This is a nice example to give when dealing with infinite series, for it shows that divergent series can be very useful if one does the right things with their partial sums. The fundamental facts about the gamma function are G(x+1)=xG(x), G(1)=1, and the integral found by Euler. A more important integral is the beta integral, studied by Wallis and evaluated in general by Euler. It is the integral of t^(x-1)(1-t)^(y-1) on [0,1], and the value is G(x)G(y)/G(x+y). The substitution t=sin^2(u) changes this form of the beta integral into one which contains the integral mentioned above, after the interval of integration above is reduced to [0,pi/2], which is the nature interval of integration for powers of sin(u). The gamma function and beta integrals (there are a few others) are fundamental functions which are building blocks of a lot of interesting and important parts of mathematics, which have applications in many areas. For example, when Robert Solow gave a talk in Madison, WI about fifteen years ago, he mentioned to me that he was the person who introduced the use of Bessel functions in economics, in connection with a circular model of a city. Bessel functions occur in many other applications. During the Second World War, Watson's book Bessel Functions was out of print, and was reprinted in 1944, when paper in England was very scarce. This 800+ page book was need for work on radar. A copy of it at the Univ. of Chicago where work was being done on the first atomic pile was chained to a table, and always open, since it was needed by many people doing work on this. The astrophysicist S. Chandrasekhar wrote a rave review of this reprint. Watson's book was first published in the early 1920s, and is still as useful now as it was in the 1940s. We know more about Bessel functions than Watson did, but what he has there is still essential for many different applications. Without the gamma and beta integrals, the study of Bessel functions would be much less rich, and much less could be done with them. While it is true that most problems can not be solved exactly, those which can are precious. Physicists use exactly solved problems to do perturbations from them to solve other problems which frequently could not be solved by other methods. As is the case with the two dimensional Ising model, solved by Onsager in the 40s, an exact solution can give real physical insight. It was only with this solution that physicists learned how to model phase transitions mathematically. Most phase trasitions are too complicated to be determined exactly, but numerical work can be done to approximate them only because of this exactly solved problem. We need to teach students both types of problems, those that can be solved exactly and those that can only be solved numerically or approximately. Anyone who only does one of these cheats the students of something very important. The problem with the solution given by the student which was described above is that he went to numerical work when he did not see any pattern in the numbers. In the newer books that I have seen, curve fitting is done for linear functions, maybe quadratic growth, and exponential growth. If the data does not fit one of these curves exactly, then there is a rush to do data fitting approximately. The data found above has a nice pattern, and the student seemed to have enough ability to have found it if he had not been programed to go to curve fitting via least squares too soon. Students need to see many different types of exactly solved problems, so that they have a large number of examples to refer to. If you want some nice problems, consider what happens to (x)_n when you take a finite difference of it, with Df(x)=f(x)-f(x-1), and try to find the inverse operator which undoes this difference operator, and see what function plays the role of x^n for both this difference operator and its inverse. The "fundamental theorem of calculus" is obvious in this setting. If you push this far enough, using the binomial theorem, you get to what is now called the Chu-Vandermonde sum, which can be written in many ways, one being
sum from 0 to n (x)_k (y)_(n-k) (x+y)_n ---------------- = --------- k! (n-k)! n!
Then it is a nice problem to see how this gives the value of the beta integral of Euler using the limit definition of the gamma function. I have done this a number of times in a first year honor section of calculus. Many of you have students taking AP calculus who are capable of learning this. If you do, you are showing the students something about the connection between the discrete and the continuous, which is very important. Derivatives and finite differences, integrals and sums go together. When people talk about having to do discrete mathematics because of computers, here is an ideal topic to do in calculus. Dick Askey email@example.com