The role of the mean value theorem (MVT) in first-year calculus

Question

Should the mean value theorem be taught in first-year calculus?

Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that interval. Most of the related exercises that books list are either about examples related to the necessity of the hypothesis of the theorem (i.e. just checking it) or about proving theoretical facts (e.g. general properties of functions, or derivatives of functions, inequalities, Taylor, ...) I can imagine a text that doesn't require the student to understand logical rigor, but that instructs them on the role of the MVT in the theory.

It is a fact that many times we fail in making students able to use MVT in these kind problems.

Assume that we are considering a first introduction to calculus for students that mostly will use it in application, students that will work in Biology, Engineering, Chemistry,...

Is it possible to remove MVT from the program and get a consistent exposition of the rest of the results and techniques in Calculus? How does eliminating MVT from the curriculum affect students from specialties exemplified above (i.e. In what ways is MVT used in further studies that they have to take or that they need)? Can the role of MVT be replaced by a more easy to use/easy to grasp result? Are there other uses (exercises) more suitable for the uses that MVT has for students of this specialties?

Note added later: "Franklin" appears to have altered the meaning of the question with his later edits. I'll say more on this soon.......

Answer 1

My view is that there are essentially two strands in a first calculus course.

The first is not really concerned with a rigorous presentation; rather it tries to get the main ideas, their interrelations, and uses across.

The second is concerned with the technicalities, showing how abstract mathematics can lead to very useful, interesting, and important results.

This means that we are really working with two different definitions of the derivative. The first is to draw the tangent line and measure its slope. The second is to compute a certain limit. To be sure, the limit is motivated by the tangent approach, but no attempt is ever made to show that the two approaches give the same answer (indeed this can't be proved using the usual definitions, since ultimately the definition of tangent line is based on the derivative).

The MVT is the basis for all proofs that geometric intuition about slopes of tangent lines holds for the limit definition. That is, the metamathematical content of the MVT is that the intuition definition matches the formal definition.

When I realized this, I decided that this point was so subtle that I either have to make a big point of explaining the question or else drop it. This choice varies from class to class.

EDIT: You use the MVT to show that positive derivative corresponds to increasing function. This is obvious from the intuitive point of view, but not from the formal point of view.

Answer 2

The purpose of this answer (which I would make CW even if the question weren't) is to collect references to scholarly articles on MVT and its role in introductory calculus courses. Most of these articles have some real mathematical content: e.g. they discuss logical implications between different forms of MVT. Certainly they are written by people who have thought deeply and in novel ways about this result -- i.e., by mathematicians.

K.A. Bush, Classroom Notes: On an Application of the Mean Value Theorem. Amer. Math. Monthly 62 (1955), 577--578, MR1529115.

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M.R. Spiegel, Mean Value Theorems and Taylor Series. Math. Mag. 29 (1956), 263--266, MR1570819.

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D. Zeitlin, Classroom Notes: An Application of the Mean Value Theorem. Amer. Math. Monthly 64 (1957), 427, MR1529634.

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C.L. Wang, Classroom Notes: Proof of the Mean Value Theorem. Amer. Math. Monthly 65 (1958), 362--364, MR1529949.

(In this article, the derivation of MVT from Rolle's theorem by tilting one's head is presented in horrible analytic detail.)

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J.P. Evans, Classroom Notes: Sequences Generated by Use of the Mean Value Theorem. Amer. Math. Monthly 68 (1961), 365, MR1531194.

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L.C. Barrett, Classroom Notes: Methods of Proving Mean Value Theorems. Amer. Math. Monthly 69 (1962), 50--52, MR1531511).

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L.W. Cohen, On being mean to the Mean Value Theorem. Amer. Math. Monthly 74 (1967), 581-582, MR1534332.

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L. Bers, On avoiding the Mean Value Theorem. Amer. Math. Monthly 74 (1967), 583, MR1534333.

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H. Levi, Classroom Notes: Integration, Anti-Differentiation and a Converse to the Mean Value Theorem. Amer. Math. Monthly 74 (1967), 585--586, MR1534334.

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R.P. Boas Jr., Classroom Notes: Lhospital's Rule Without Mean-Value Theorems. Amer. Math. Monthly 76 (1969), 1051--1053, MR1535647.

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D.E. Sanderson, Classroom Notes: A Versatile Vector Mean Value Theorem. Amer. Math. Monthly 79 (1972), 381--383, MR1536685.

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R.P. Boas Jr., Who needs these mean-value theorems anyway? Two-Year College Math. J. 12 (1981), 178--181.

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T.W. Tucker, Rethinking Rigor in Calculus: The Role of the Mean Value Theorem. Amer. Math. Monthly 104 (1997), 231--240, MR1436045, MAA website.

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H. Swann, Commentary on Rethinking Rigor in Calculus: The Role of the Mean Value Theorem. Amer. Math. Monthly 104 (1997), 241--245, MR1436046.

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J.J. Koliha, Mean, Meaner, and the Meanest Mean Value Theorem. Amer. Math. Monthly 116 (2009), 356--361, MR2503322, JSTOR.

This list was compiled as follows: first I started with the five or so articles on MVT that I remembered, mostly from having appeared in the Monthly. Then I did a MathSciNet search: "Anywhere=(mean value theorem)". This gives 1589 matches. Starting with the earliest papers, I looked quickly through the articles until I got tired. My list contains all the articles that I spotted and thought to be of relevance to the question which were published before 1973. This was about 25% of the 1589 matches.

Answer 3

I think your criticisms of calculus textbooks are on the mark.

In my view, if you are teaching students how to use calculus and not how to prove every assertion in the subject, there is no reason to state or use the Mean Value Theorem. If you must state it, the visual "proof" is best. Some claim it is needed for deriving the error term for a Taylor polynomial, but in my view the integral representation of the error is far more useful and easily derived from the Fundamental Theorem of Calculus.

(Wandering off topic here) The standard calculus textbook is, in my opinion, a confused logically inconsistent combination of rigorous reasoning and take-it-on-faith assertions. For example, a lot of textbooks devote a lot of attention to showing how to define $e$ and the exponential function rigorously (to the point that some textbooks find it necessary to define the logarithm first!). On the other hand, I have never seen any textbook worry about how to define degrees and radians rigorously.

To elaborate a little, many uses of the MVT:

There exists $c \in [a, b]$ such that $f(b) - f(a) = f'(c)(b-a)$

can be achieved just as well by the fundamental theorem of calculus:

$f(b) - f(a) = A(b-a),$

where

$A = \frac{1}{b-a}\int_a^b f'(t)\,dt$

is the average derivative of $f$ on the interval $[a,b]$.

Answer 4

Although this topic may not contain research level mathematics, perhaps the perspective of a researcher is useful in creating new ways of presenting it. After many years of teaching the MVT is various ways unsuccessfully, I came to consider it more useful and practical to try to convey the corollaries of the MVT rather than the statement. If the statement is taught I made it an assertion in words rather than just a formula. I.e. my students struggled unsuccessfully to recall whether the statement contained $f(b)-f(a)$ or $f(a)-f(b)$, whereas I really wanted them to be able to apply it to solve antiderivative problems. Even those students who got the formula right had trouble deducing its meaning. It was also a challenge for students to use inequalities to find intervals on which the derivative was always positive. Hence I developed a presentation, including all proofs if desired, that do not involve manipulating formulas or inequalities. In particular, one consequence is that even the statement of the MVT is not needed at all until one does antiderivative problems.

I agree that what and how to teach depend on ones goals. The narrow goals I set for my first semester course are for students to learn four basic principles, stated in words, and their simplest applications.

1) (“Intermediate value theorem”) the continuous image of an interval is again an interval, applied to existence of solutions of equations.

2)(“Max min value theorem”) the continuous image of a closed bounded interval is again closed and bounded, with applications to existence of solutions of closed interval max min problems.

3)(“Rolle theorem”) A differentiable function is monotone on an interval not containing a critical point, applied to graphing problems and open interval max min problems.

4) (Cor. of MVT) Two differentiable functions with the same derivative in an interval differ there by a constant, with applications to problems such as finding areas by antiderivatives.

Only the last statement requires the mean value theorem, and that only in the proof. In particular I consider statement 4) more important for students to know than the usual statement of the MVT. The deep results, involving completeness, are the first two. Principle 3), including the easy corollary that a function increases on intervals where its derivative is positive, requires only the Rolle theorem. Here is a possible presentation along those lines.

The first two principles, the intermediate value theorem and the max min theorem are probably best assumed in a non honors course, but can be proved without least upper bounds, by constructing recursively an infinite decimal that satisfies the desired theorem. Using 1), the third principle is equivalent to Rolle’s theorem. I.e. Rolle says a differentiable function taking the same value twice has derivative zero in between, and by principle 1) any function that changes direction takes the same value twice. Hence a differentiable function is strictly monotone on any interval not containing a critical point, and it follows easily that it increases where $f’>0$. It also follows easily that a function whose derivative changes sign changes direction hence has a critical point, so the derivative cannot change sign without becoming zero, i.e. derivatives have the intermediate value property. Thus checking monotonicity requires only finding, usually finitely many critical points, and evaluating the derivative at one point between successive critical points, not on a whole interval. It is significantly easier for students to apply this finitistic process of graphing. The MVT is not needed for any of this.

If one wants to prove principle 4), the relevant statement equivalent to MVT is that two differentiable functions $f,g$ that agree with each other at two points must also have the same derivative somewhere in between. This of course is immediate from Rolle applied to $f-g$. Then since every function $f$ agrees at any two points $a,b$ with some linear function (the secant), and the only linear function with derivative zero is constant, it follows that a function with derivative zero everywhere agrees at any two points with a constant function, hence is itself constant. This implies 4) as usual by subtraction.

This suggests, to me at least, that the formula in the usual MVT is only a needlessly belabored restatement of the numerical slope of a line, which adds nothing to the usefulness of the theorem. I do not argue that anyone else should agree with these ideas. I mention them as a suggestion for those who wish to minimize the use of MVT for some of its usual applications. I.e. the central and deep result is the max min value theorem and the basic fact that an interior extremum is a critical point, as noted above, and the easy corollary stated in the Rolle theorem suffices for all applications short of antiderivative problems. I admit that in a later course where Taylor series enter, the explicit formula begins to look useful.

Answer 5

I have just been teaching calculus for the first time, and I am firmly of the opinion that in many calculus courses, the mean value theorem should have essentially no role. All the applications of it can be explained intuitively without any reference to it, and the semblance of "rigor" that using it provides is largely obscured by the fact that it is presented as merely a black box without a proof of its own. It's a fairly arbitrary principle from which to rigorously derive basic facts of calculus, and if we were to provide completely rigorous proofs of everything, there are other, more intuitive proofs that do not use MVT (typically using instead a compactness argument, the technicalities of which can easily be brushed aside while leaving the key idea of the proof clear).

That said, this impression is based on my one experience with a particular course which talked about MVT but only quite briefly. I could imagine it fitting well into a different course which was overall much more rigorous and which went through Rolle's Theorem and MVT in greater depth.

Answer 6

I believe students must be guided by intuition in a first year calculus course.

Therefore, it is better to use geometry to explain derivatives in terms of tangent lines and geometric limits of pushing points together. A student cannot appreciate the need for rigor if their intuition is not first sufficiently developed.

Since the MVT is mostly useful for rigorous computations, notably for the proof of the fundamental theorem of calculus, I agree that it is out of place in an ideal introduction to calculus.

My main objection is that it is a long side track to prove, and it seems pointless to a student who doesn't appreciate rigor, which is likely to turn them off to mathematics.

Furthermore, The fundamental theorem can be argued for intuitively by discussing the geometric ideas behind integration. When the student is able to see the flaws in the intuitive argument, they will also be able to appreciate the MVT.

Answer 7

First I'll apologize for necrobumping this thread after well over a year, but, it seemed to me that there is a simple answer to the OP's question which was never directly addressed.

Students are interested in existence and uniqueness of solutions to equations. They like the fact that $x^2 = -1$ has no solutions (in the real numbers) but $x^2=1$ has two solutions. They like the fact that $2x + 5y = 1$ has infinitely many solutions (in the $(x,y)$-coordinate plane).

I like to ask students: what are the solutions of the equation $y'=1$? The format of the answer must be a function $y=f(x)$. And of course they can all come up with infinitely many different solutions.

Then I ask them: are there any other solutions? If not, why not? They don't have to be "interested" in abstract mathematics and proofs and rigor in order to appreciate this question and to take interest when I show them why the functions $x + C$ are, in fact, all the solutions. And then one easily continues on to solutions of equations like $y' = $ any other of your favorite functions.

So yes, MVT can and should be taught in calculus.

Answer 8

In my opinion it should be taught - but it is very important to give an intuition. A very nice exposition how main ideas of the calculus could be connected in a way an undergraduate should understand can be found here:

http://web.ncf.ca/fs039/mp/documents/mftc.pdf

Answer 9

I agree there are many problems in the approaches done in many of the calculus books used but I disagree about the mean value theorem (Lagrange theorem for me). It is the cornerstone of analysis. You probably have some treatment in mind or a whole list of them. Lagrange theorem have the combined power of Bolzano's theorem (continuity of the reals, for what is worth) and the notion of derivative. If you want to pass global info from the derivative to the function, the mean value theorem is the place to go. Of course one has to be clear that the problem is really about the "how" are things presented. After all MVT is equivalent to the continuity of the reals.

One trivial change that I always try to do is a simple change in the writing. Instead of writing the equation with the derivative isolated in one side write the function isolated (like a Taylor). Also with the definition of derivative. instead of writing the derivative in one side of the equation writing it inside the limit. That apparently unimportant change has as outcome that students grasp better the connection between them: generalize MVT to Taylor, derivative to differential, use MVT in application.

--more added after the first comment---

Oh, that's true. Nevertheless the two are related. Of course the question about teaching is not a well stated one. It depends on the goals of the course, i.e. what it is that you want your students to be able to do. One unavoidable goal in a calculus course is to study nice functions. Nowadays, this means continuous and differentiable functions (although a close look at most of the courses tells us that the class of interest is much narrower.)

The concept of derivative seems to be then, required, although I bet a good course can also be planned with the notion of power series instead (which seems to be returning to times before Newton-Leibntz but I am not so sure [ask Doron Zeilberger about it]). A less chocking approach is to put both concepts side by side. And MVT is a way of linking them link.

I have to say that the way programs evolve is by taking the old ones and doing little "improvements". It is true that in the scope of basic ordinary calculus course you can skip MVT only losing the possibility of asking problems like "prove that $|sin(x)|\leq|x|$".

But again, it is a problem of goals. It is also good to take into account that learning process works starting from the horizon of already acquired knowledge. Even if students at the end cannot even remember the statement or how to use it, it prepares them for further development. Or just remember, you the working mathematician, how many times (if not every time), you have gone to a conference in which you don't understand a thing, in which you only remember two or three names at the end. Now remember how many times just knowing that that name (or word) exists or that is related to some other name has opened a completely new road in your research. It works exactly the same for students, even though they are not doing research. I say this to point out that the validity of an element in a curriculum program should not only be judged by the ability of students to actually get it but also by the grounds it creates to build on top of it. Education is a process that involve not only teaching but also evaluating. Maybe it is better to look at how MVT is evaluated instead. If it is playing a role of a connective element then it is wrong to evaluate the skills of applying it (which involve both understanding and skill). Changing the evaluation method is a less destructive approach than elimination from the program. Uff, I have written too much. If I forgot to say something I will say it later.

Answer 10

As no one seems to have mentioned it, students in the specialties listed in the OP frequently need to understand the behaviour of solutions of ordinary differential equations. I haven't taught such a course recently enough to crank out a list of places where it comes up, but abstractly I feel like there are a "zillion" situations where you end up using MVT to justify some basic intuition about solutions of ODE. For example, "The vector field points down and left over here, so once a solution enters region A, it can never leave", or certain kinds of stability analysis (e.g. when linear stability analysis fails).

Answer 11

In my opinion (I have been a TA for 4 years), MVTs should be given in the following forms: MVT-Integrals: [Diestel] (Soft version) $f$ is B-integrable. Then for any $|E|>0,$ $\frac{\int_E fd\mu}{|E|}\in \bar{co}(f(E))$

MVT and MVT-Cauchy: As given by Apostol in his book Calc. vol I with convexity

Function theory heavily depends on these theorems+FTC. MVTs are related to, for instance, complex analysis which is a must for an engineer. Definitions of holomorphic functions and (sub)harmonic functions contain both volume and surface integral averages. Details can be found in [Krantz]. Also Taylor series(both real,complex analytic cases) uses these theorems. I should also mention the `''convexity'' which is very fundamental.