Anton, Bivens, and Davis have written calculus books with late

transcendentals and Stewart has a calculus book with early transcendentals.

What's this all about?

edit 1: (Both terms show up in the titles of the books.)

Here's the Anton, Bivens, Davis book:

Here's a link to an amazon page selling the Stewart calculus book:

  • 1
    I'm not sure what you mean by this. Can you clarify, perhaps with some relevant quotations from the books? – Mike Miller Oct 22 '13 at 4:05
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    Stewart's book introduces transcendental functions early in the book (ch.1), whereas the other book postpones them until later in the book (ch.6). This isn't a mathematical question. – David H Oct 22 '13 at 4:13
  • 1
    People in Chemistry, Physics, Engineering need their students to be quickly comfortable with the exponential function, the logarithm, and the trig functions. – André Nicolas Oct 22 '13 at 4:15
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    So the the difference between "early trans" and "late trans" is a difference between the outlines of the book, and not the material? This is not about "different kinds of transcendental numbers or transcendental functions"? – ninnymonger Oct 22 '13 at 4:18
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    I think the question is interesting, on-topic (clearly not on the off-topic list even if not in the welcome list) and should have an answer that is flagged as such. Instead of taking the time to close it, take the time to answer it and let it be. Googled early transcendentals and came here. – Ludovic Kuty Aug 11 '14 at 19:01
up vote 35 down vote accepted

The difference is entirely a pedagogical one. Both approaches ultimately cover the same calculus.

“Late transcendentals” is the traditional approach to teaching calculus where the treatment of logarithmic and exponential functions is postponed until after integration is introduced. In the traditional method, the natural logarithm is defined by the formula $$ \ln x \;\underset{\scriptscriptstyle\mathrm{def}}{=}\; \int_1^x \frac{1}{t}\,dt, $$ and $e^x$ is then defined to be the inverse function of $\ln x$. Important properties of $\ln x$, such as that $\ln(xy) = \ln x + \ln y$, are proven from the integral definition using $u$-substitution, and the properties of the exponential function follow from these. Arbitrary powers $a^b$ where $a>0$ and $b\in\mathbb{R}$ are then defined by the formula $$ a^b = e^{b \ln a} $$ and it is shown that this agrees with the existing definition when $b$ is rational.

In the “early transcendentals” method, the logarithmic and exponential functions are introduced shortly after the definition of the derivative. Exponentiation where the power is an arbitrary real number is defined by the formula $$ a^b \;=\; \lim_{q\to b}\, a^q $$ where the limit is taken over the rational numbers. (There are some technical issues in proving that this limit exists, which are often ignored.) An argument is then given that there exists a number $e$ for which $$ \frac{d}{dx}\bigl[e^x\bigr]\biggr|_{x=0} \;=\; 1, $$ though again the presentation of this argument is sometimes less than completely rigorous. It follows that the derivative of $e^x$ is $e^x$, and the natural logarithm is defined as the inverse of the exponential function.

The traditional method has the advantage of being cleaner, and the proofs are simple enough that they can be presented to starting calculus students in a mathematically rigorous way. However, it has several distinct disadvantages:

  1. It is not very intuitive, since the natural logarithm and exponential function are essentially summoned out of thin air by what looks like magic. The “early transcendentals” approach, on the other hand, corresponds much more closely with how we actually think of exponentials and logarithms.

  2. Students who take only a single semester of calculus, which includes most biology majors at some colleges in the United States, do not see the exponential function and natural logarithm. This is a serious problem, because these are among the most important functions for applications in biology.

  3. Even students who take two semesters of calculus learn about exponential and logarithmic functions fairly late, which means they don't have time to get used to these functions.

These arguments are generally considered persuasive enough that most universities in the United States have now adopted calculus books that use the “early transcendentals” approach.

Of course, not everyone agrees with this change, and further arguments can be presented on both sides of the debate. Books using the traditional approach continue to be used at some universities and in some calculus classes. In particular, some universities offer an honors calculus sequence designed specifically for math majors and similar students, and there's a relatively strong argument for using the traditional approach in such a course. My impression is also that the “late transcendentals” approach remains quite popular outside the United States, though my evidence for this is purely anecdotal.

  • 1
    I think the "early transcendentals" approach probably makes more sense from a foundational perspective, since $\mathbb{C}$ has an exponential function but this doesn't have a single-valued inverse. Therefore I'd be inclined to say that exponentials are the more fundamental concept. You've also got the notion of an exponential map from Lie theory. – goblin Jul 16 '15 at 14:02

I studied Calculus using the "early transcendentals" version of Anton [look this link of the book] 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".

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