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Jack Adams
Jack Adams

Calculus: Early Transcendental Functions

Most American universities seem to use the early transcendentals approach, but I have never seen any empirical evidence that it is superior (e.g. higher pass rates, success rates in courses which depend on calculus). So my question is:

Calculus: Early Transcendental Functions

However I believe it's important to use these functions as much as possible and in as many different combinations from as early as possible (we do so at our college as soon as we start second year work), precisely because they behave very differently to each other and to polynomials, so that the students have a better understanding of calculus which they would otherwise perceive as primarily about powers and coefficients.

I believe early diversity is helpful in correctly generalising concepts from examples to principles, and the longer they are exposed to these beautiful and interesting functions the better they will be able to deal with them.

And so there will be picking and choosing when it comes to which functions are to be introduced, and to which derivatives or antiderivatives are to be taken. Given the nature of the AP examinations and the subsequent courses in Calculus in most mathematics departments, I think it wise to introduce at least the functions mentioned in what you call the early approach.

Many places have Calculus I as a co-requisite for University Physics I. The physics instructors like students in thier class to be familiar with derivatives of exponential functions before the end of the semester, hence a need for early transcendentals.

The late "transcendentals approach" has the huge drawback, that students often miss out the mere algebraic properties (like linearity) of these functions. And in all but the simplest calculus examples involving transcendental functions, these algebraic properties are important as well.

I know that this is an old question, but it should be pointed out that in the United States at least, "early transcendentals" vs. "late transcendentals" refers to the placement of exponential, logarithmic, and inverse trigonometric functions. I've never seen a textbook which didn't introduce the derivatives of trigonometric functions in the first semester, but I have seen a number of texts which delay exponential and logarithmic functions until the second semester. As a simple example -- compare the table of contents of James Stewart's calculus texts. The one with "Early Transcendentals" in its title covers logarithms and exponential functions before integration and the one without that in the title covers them just after integration is introduced. Both cover trigonometric functions just after polynomials.

Where I teach we use the early transcendentals approach. I was resistant at first since I had both originally learned and previously taught using the other approach, but I now see the advantages of the earlier approach. The chief advantage is that it is better for the science and engineering students who (at least where I teach) often make up the majority of a first semester calculus class. In particular, our biology majors are only required to take a single semester of calculus and it would be a shame if they ended their calculus without seeing the link to exponential growth and decay.

I see two disadvantages to early transcendentals. One is that it pushes back integration to the very end of the first semester. I am chronically pressed for time towards the end of the first semester. I am able to get through the Fundamental Theorem of Calculus, but often at the cost of being very hand-wavy about things like Riemann sums. The early transcendentals tends to force me to cover integration more superficially than I would like.

A second disadvantage is that it is almost impossible to give a rigorous approach to exponential and logarithms without knowing about the Fundamental Theorem of Calculus. I don't know of any adequate definition of a^x for x irrational that can be shown to make sense using only the tools of early calculus. At best you can say something like if a^x exists and if certain limits exist, then it is a differentiable function. The entire discussion is an almost hopeless hand-wave. In contrast, with the late transcendentals approach you can define ln(x) as the definite integral of 1/t from 1 to x and work out the properties of both logarithms and exponential functions from there. This is clearly a better approach for pure math majors. In addition to being more rigorous, it leaves them with a better understanding of what the Fundamental Theorem actually means.

After teaching calculus for a long time, I have found that actually doing math during class is absolutely essential. By this I mean that you should justify as much as you possibly can, and tell a story about how these ideas arise. The other major way I have seen math taught is through a slide show where concepts are listed and then applied, as if they are magically true and the only interesting thing to do in math is applying these concepts to specific problems. When math is presented as a sequence of concepts that are applied to solve problems, students do not experience math as a coherent language that itself leads to new concepts derived from familiar ones. Under this approach, I cannot define ln(x) until one can integrate functions, knows the mean value theorem, and of course can use limits. If f(x)=ln(x) is introduced before integrals, how do you define it? As the inverse to g(x)=e^x? OK, what is the definition of e^x, and how do derivatives and inverses relate? By going down the path of early transcendentals, you can find more derivatives earlier in your college career (by 2 months or so), but you can't integrate until later in your college career, and also don't know the definitions of many of the functions that you are using, nor how they are related.

I studied Calculus using the "early transcendentals" version of Anton [look this link of the book -Transcendentals-Combined-Howard-Anton/dp/0471472441] and I remember that the book explain natural logarithms without integrals, and after, explain again natural logarithms using the integral notation. I liked the "early transcendentals" method, because ln(x) = log(e, x), that means, natural logarithm can be understand like a logarithm in the special base "e". 041b061a72


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