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Exponential functions have the form bx, where b > 0 is the base and x the exponent.

Exponential functions are easily introduced as growth processes. The comparison of x² and 2^x is an eye-opener, with the stories of duckweed or the grain on the chess board. The introduction of the exponential number e is a next step. What intuitions can we use for smooth didactics on e ?

The “discover-e” plot

There is the following “intuitive graph” for the exponential number e = 2,71828…. The line y = e is found by requiring that the inclines (tangents) to bx all run through the origin at {0, 0}. The (dashed) value at x = 1 helps to identify the function ex itself. (Check that the red curve indicates 2^x).

Functions 2^x, e^x and 4^x, and tangents through {0, 0}

2^x, e^x and 4^x, and inclines through {0, 0}

Remarkably, Michael Range (2016:xxix) also looks at such an outcome = 2^(1 / c), where is the derivative of = 2^x at x = 0, or c = ln[2]. NB. Instead of the opaque term “logarithm” let us use “recovered exponent”, denoted as rex[y].

Perhaps above plot captures a good intuition of the exponential number ? I am not convinced yet but find that it deserves a fair chance.

NB. Dutch mathematics didactician Hessel Pot, in an email to me of April 7 2013, suggested above plot. There appears to be a Wolfram Demonstrations Project item on this too. Their reference is to Helen Skala, “A discover-e,” The College Mathematics Journal, 28(2), 1997 pp. 128–129 (Jstor), and it has been included in the “Calculus Collection” (2010).

Deductions

The point-slope version of the incline (tangent) of function f[x] at x = a is:

y – f[a] = s (x a)

The function b^x has derivative rex[b] b^x. Thus at arbitrary a:

y – b^a = rex[b] b^a (x a)

This line runs through the origin {xy} = {0, 0} iff

0 – b^a = rex[b] b^a (0 – a)

1 = rex[ba

Thus with H = -1, a = rex[b]H = 1 / rex[b]. Then also:

yf[a] = b^a = b^rex[b]H = e^(rex[b]  rex[b]H) = e^1 = e

The inclines running through {0, 0} also run through {rex[b]H, e}. Alternatively put, inclines can thus run through the origin and then cut y = e .

For example, in above plot, with 2^x as the red curve, rex[2] ≈ 0.70 and ≈ 1.44, and there we find the intersection with the line y = e.

Subsequently also at a = 1, the point of tangency is {1, e}, and we find with e that rex[e] = 1,

The drawback of this exposition is that it presupposes some algebra on e and the recovered exponents. Without this deduction, it is not guaranteed that above plot is correct. It might be a delusion. Yet since the plot is correct, we may present it to students, and it generates a sense of wonder what this special number e is. Thus it still is possible to make the plot and then begin to develop the required math.

Another drawback of this plot is that it compares different exponential functions and doesn’t focus on the key property of e^x, namely that it is its own derivative. A comparison of different exponential functions is useful, yet for what purpose exactly ?

Descartes

Our recent weblog text discussed how Cartesius used Euclid’s criterion of tangency of circle and line to determine inclines to curves. The following plots use this idea for e^x at point x = a, for a = 0 and a = 1.

Incline to e^x at x = 0 (left) and x = 1 (right)

Incline to e^x at x = 0 (left) and x = 1 (right)

Let us now define the number e such that the derivative of e^x is given by e^x itself. At point x = a we have s = e^a. Using the point-slope equation for the incline:

y – f[a] = s (x a)

y – e^ae^a (x a)

y e^a (x – (a – 1))

Thus the inclines cut the horizontal axis at {x, y} = {a – 1, 0}, and the slope indeed is given by the tangent s = (f[a] – 0) / (a – (a – 1)) = f[a] / 1 = e^a.

The center {u, 0} and radius r of the circle can be found from the formulas of the mentioned weblog entry (or Pythagoras), and check e.g. a = 0:

u = a + s f[a] = a + (e^a

r = f[a] √ (1 + s²) = e^a √ (1 + (e^a)²)

A key problem with this approach is that the notion of “derivative” is not defined yet. We might plug in any number, say e^2 = 10 and e^3 = 11. For any location the Pythagorean Theorem allows us to create a circle. The notion of a circle is not essential here (yet). But it is nice to see how Cartesius might have done it, if he had had e = 2.71828….

Conquest of the Plane (COTP) (2011)

Conquest of the Plane (2011:167+), pdf online, has the following approach:

  • §12.1.1 has the intuition of the “fixed point” that the derivative of e^x is given by e^x itself. For didactics it is important to have this property firmly established in the minds of the students, since they tend to forget this. This might be achieved perhaps in other ways too, but COTP has opted for the notion of a fixed point. The discussion is “hand waiving” and not intended as a real development of fixed points or theory of function spaces.
  • §12.1.2 defines e with some key properties. It holds by definition that the derivative of e^x is given by e^x itself, but there are also some direct implications, like the slope of 1 at x = 0. Observe that COTP handles integral and derivative consistently as interdependent notions. (Shen & Lin (2014) use this approach too.)
  • §12.1.3 gives the existence proof. With the mentioned properties, such a number and function appears to exist. This compares e^x with other exponential functions b^x and the recovered exponents rex[y] – i.e. logarithm ln[y].
  • §12.1.4 uses the chain rule to find the derivatives of b^x in general. The plot suggested by Hessel Pot above would be a welcome addition to confirm this deduction and extension of the existence proof.
  • §12.1.5-7 have some relevant aspects that need not concern us here.
  • §12.1.8.1 shows that the definition is consistent with the earlier formal definition of a derivative. Application of that definition doesn’t generate an inconsistency. No limits are required.
  • §12.1.8.2 gives the numerical development of = 2.71828… There is a clear distinction between deduction that such a number exists and the calculation of its value. (The approach with limits might confuse these aspects.)
  • §12.1.8.3 shows that also the notion of the dynamic quotient (COTP p57)  is consistent with above approach to e. Thus, the above hasn’t used the dynamic quotient. Using it, we can derive that 1 = {(e^h – 1) // h, set h = 0}. Thus the latter expression cannot be simplified further but we don’t need to do so since we can determine that its value is 1. If we would wish so, we could use this (deduced) property to define e as well (“the formal approach”).

The key difference between COTP and above “approach of Cartesius” is that COTP shows how the (common) numerical development of e can be found. This method relies on the formula of the derivative, which Cartesius didn’t have (or didn’t want to adopt from Fermat).

Difference of COTP and a textbook introduction of e

In my email of March 27 2013 to Hessel Pot I explained how COTP differed from a particular Dutch textbook on the introduction of e.

  • The textbook suggests that f ‘[0] = 1 would be an intuitive criterion. This is only partly true.
  • It proceeds in reworking f ‘[0] = 1 into a more general formula. (I didn’t mention unstated assumptions in 2013.)
  • It eventually boils down to indeed positing that e^x has itself as its derivative, but this definition thus is not explicitly presented as a definition. The clarity of positing this is obscured by the path leading there. Thus, I feel that the approach in COTP is a small but actually key innovation to explicitly define e^x as being equal to its derivative.
  • It presents e only with three decimals.
Conclusion

There are more ways to address the intuition for the exponential number, like the growth process or the surface area under 1 / x. Yet the above approaches are more fitting for the algebraic approach. Of these, COTP has a development that is strong and appealing. The plots by Cartesius and Pot are useful and supportive but no alternatives.

The Appendix contains a deduction that was done in the course of writing this weblog entry. It seems useful to include it, but it is not key to above argument.

Appendix. Using the general formula on factor x a

The earlier weblog entry on Cartesius and Fermat used a circle and generated a “general formula” on a factor x a. This is not really factoring, since the factor only holds when the curve lies on a circle.

Using the two relations:

f[x] – f[a]  = (x a)  (2u – x – a) / (f[x] + f[a])    … (* general)

u = a + s f[a]       … (for a tangent to a circle)

we can restate the earlier theorem that s defined in this manner generates the slope that is tangent to a circle. 

f[x] – f[a]  = (x a)  (2 s f[a](x – a)) / (f[x] + f[a]) 

It will be useful to switch to x a = h:

f[a + h] – f[a]  = h (2 s f[a] – h) / (f[a + h] + f[a]) 

Thus with the definition of the derivative via the dynamic quotient we have:

df / dx = {Δf // Δx, set Δx = 0}

= {(f[a + h] – f[a]) // h, set h = 0}

= { (2 s f[a] – h) / (f[a + h] + f[a]), set h = 0}

= s

This merely shows that the dynamic quotient restates the earlier theorem on the tangency of a line and circle for a curve.

This holds for any function and thus also for the exponential function. Now we have s = e^a by definition. For e^x this gives:

ea + hea  = h (2 s eah) / (ea + h + ea)

For COTP §12.1.8.3 we get, with Δx = h:

df / dx = {Δf // Δx, set Δx = 0}

= {(ea + hea  ) // h, set h = 0}

= {(2 s eah) / (ea + h + ea) , set h = 0}

= s

This replaces Δf // Δx by the expression from the general formula, while the general formula was found by assuming a tangent circle, with s as the slope of the incline. There is the tricky aspect that we might choose any value of s as long as it satisfies u = a + s f[a]. However, we can refer to the earlier discussion in §12.1.8.2 on the actual calculation.

The basic conclusion is that this “general formula” enhances the consistency of §12.1.8.3. The deduction however is not needed, since we have §12.1.8.1, but it is useful to see that this new elaboration doesn’t generate an inconsistency. In a way this new elaboration is distractive, since the conclusion that 1 = {(e^h – 1) // h, set h = 0} is much stronger.

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Isaac Newton (1642-1727) invented the differentials, calling them evanescent quantities. Since then, the world has been wondering what these are. Just to be sure, Newton wrote his Principia (1687) by using the methods of Euclidean geometry, so that his results could be accepted in the standard of his day (context of reconstruction and presentation), and so that his results were not lost in a discussion about the new method of these differentials (context of discovery). However, this only increased the enigma. What can these quantities be, that are so efficient for science, and that actually disappear when mathematically interesting ?

Gottfried Leibniz (1646-1716) gave these infinitesimals their common labels dy and dx, and thus they became familiar as household names in academic circles, but this didn’t reduce their mystery.

Charles Dodgson (1832-1898) as Lewis Carroll had great fun with the Cheshire Cat, who disappears but leaves its grin.

Abraham Robinson (1918-1974) presented an interpretation called “non-standard analysis“. Many people think that he clinched it, but when I start reading then my intuition warns me that this is making things more difficult. (Perhaps I should read more though.)

In 2007, I developed an algebraic approach to the derivative. This was in the book “A Logic of Exceptions” (ALOE), later also included in “Elegance with Substance” (EWS) (2009, 2015), and a bit later there was a “proof of concept” in “Conquest of the Plane” (COTP) (2011). The pdfs are online, and a recent overview article is here. A recent supplement is the discussion on continuity.

In this new algebraic approach there wasn’t a role for differentials, yet. The notation dy / dx = f ‘[x] for y f [x] can be used to link up to the literature, but up to now there was no meaning attached to the symbolism. In my perception this was (a bit of) a pity since the notation with differentials can be useful on occasion, see the example below.

Last month, reading Joop van Dormolen (1970) on the didactics of derivatives and the differential calculus – in a book for teachers Wansink (1970) volume III – I was struck by his admonition (p213) that dy / dx really is a quotient of two differentials, and that a teacher should avoid identifying it as a single symbol and as the definition of the derivative. However, when he proceeded, I was disappointed, since his treatment didn’t give the clarity that I looked for. In fact, his treatment is quite in line with that of Murray Spiegel (1962), “Advanced calculus (Metric edition)”, Schaum’s outline series, see below. (But Van Dormolen very usefully discusses the didactic questions, that Spiegel doesn’t look into.)

Thus, I developed an interpretation of my own. In my impression this finally gives the clarity that people have been looking for starting with Newton. At least: I am satisfied, and you may check whether you are too.

I don’t want to repeat myself too much, and thus I assume that you read up on the algebraic approach to the derivative in case of questions. (A good place to start is the recent overview.)

Ray through an origin

Let us first consider a ray through the origin, with horizontal axis x and vertical axis y. The ray makes an angle α with the horizontal axis. The ray can be represented by a function as y =  f [x] = s x, with the slope s = tan[α]. Observe that there is no constant term (c = 0).

2016-12-08-ray

The quotient y / x is defined everywhere, with the outcome s, except at the point x = 0, where we get an expression 0 / 0. This is quite curious. We tend to regard y / x as the slope (there is no constant term), and at x = 0 the line has that slope too, but we seem unable to say so.

There are at least three responses:

(i) Standard mathematics then takes off, with limits and continuity.

(ii) A quick fix might be to try to define a separate function to find the slope of a ray, but we can wonder whether this is all nice and proper, since we can only state the value s at 0 when we have solved the value elsewhere. If we substitute y when it isn’t a ray, or example x², then we get a curious construction, and thus the definition isn’t quite complete since there ought to be a test on being a ray.

2016-12-10-slopeofray

 

 

(iii) The algebraic approach uses the following definition of the dynamic quotient:

y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }

Thus in this case we can use y // x = s x // x = s, and this slope also holds for the value x = 0, since this has now been included in the domain too.

In a nutshell for dy / dx

In a nutshell, we get the following situation for dy / dx:

2016-12-08-dydx

Properties are exactly as Van Dormolen explained:

  • “dy” and “dx” are names for variables, and thus they have their own realm with their own axes.
  • The definition of their relationship is dy = f ‘[x] dx.

The news is:

  • The mistake in history was to write dy / dx instead of dy // dx.

The latter “mistake” can be understood, since the algebraic approach uses notions of set theory, domain and range, and dynamics as in computer algebra, and thus we can forgive Newton for not getting there yet.

To link up with history, we might define that the “symbol dy / dx as a whole” is a shortcut for dy // dx. This causes additional yards to develop the notion of “symbol as a whole” however. My impression is that it is better to use dy // dx unless it is so accepted that it might become pedantic. (You must only explain that the Earth isn’t flat while people don’t know that yet.)

Application to Spiegel 1962 gives clarity

Let us look at Spiegel (1962) p58-59, and see how above discussion can bring clarity. The key points can all be discussed with reference to his figure 4-1.

1962-murrayspiegel-p58-fig4-1

Looking at this with a critical eye, we find:

  • At the point P, there is actually the creation of two new sets of axes, namely, both the {Δx, Δy} plane and the {dx, dy} plane.
  • These two new planes have both rays through the origin, one with angle θ and one with angle α.
  • The two planes help to define the error. An error is commonly defined from the relation “true value = estimate + error”. The true value of the angle is θ and our estimate is α.
  • Thus we get absolute error Δf = s Δx + ε where s = dy / dx. This error is a function of Δx, or ε = ε[Δx]. It solves as ε = Δf – s Δx.
  • The relative error is Δf / Δx =  dy / dx + r which solves as r = Δf / Δx – dy / dx. This is still a function rx]. We use the quotient of the differentials instead of the true quotient of the differences.
  • We better re-consider the error in terms of the dynamic quotient, replacing / by // in the above, because at P we like the error to be zero. Thus in above figure we have ε = Δf  s Δx, where s = dy // dx.
  • A source of confusion is that Spiegel suggests that d≈ Δx or even dx = Δx but this is numerically true only sometimes and conceptually there surely is no identity since these are different axes.
  • In the algebraic approach, Δx is set to zero to create the derivative, in particular the value of f ‘[x] = tan[α] at point P.  In this situation, Δx = 0 thus clearly differs from the values of dx that are still available on dx ‘s whole own axis. This explains why the creation of the differentials is useful. For, while Δx is set to 0, then the differentials can take any value, including 0.

Just to be sure, the algebraic approach uses this definition:

f ’[x] = {Δf // Δx, then set Δx = 0}

Subsequently, we define dy = f ‘[x] dx, so that we can discuss the relative error r = Δf // Δx – dy // dx.

PM. Check COTP p224 for the discussion of (relative) error, with the same notation. This present discussion still replaces the statement on differentials in COTP p155, step number 10.

A subsequent point w.r.t. the standard approach

Our main point thus is that the mistake in history was to write dy / dx instead of dy // dx. There arises a subsequent point of didactics. When you have real variables and z, then these have their own axes, and you don’t put them on the same axis just because they are both reals.

See Appendix A for a quote from Spiegel (1962), and check that it is convoluted at times.

Appendix B contains a quote from p236 from Adams & Essex (2013). We can see the same confusions as in Spiegel (1962). It really is a standard approach, and convoluted.

The standard approach takes Δx = dx and joins the axis for the variable Δy with the axis for the variable dy, with the common idea of “a change from y“. The idea of this setup is that it shows the error for values of Δx = dx.

2016-12-10-delta-and-d

It remains an awkward setup. It may well be true that John from Los Angeles is called Harry in New York, but when John calls his mother back home and introduces himself as “Mom, this is Harry”, then she will be confused. Eventually she can get used to this kind of phonecalls, but it remains awkward didactics to introduce students to these new concepts in this manner. (Especially when John adds: “Mom, actually I don’t exist anymore because I have been set to zero.”)

Thus, in good didactics we should drop this Δx = dx.

Alternatively put: We might define dy = f ’[x] Δx = f // Δx, then set Δx = 0} Δx. In the latter expression Δx occurs twice: both as a local and bound variable within { … } and as a global free variable outside of { … }. This is okay. In the past, mathematicians apparently thought that it might make things clearer to write dfor the free global variable: dy = f ’[x] dx. In a way this is okay too. But for didactics it doesn’t work. We should rather avoid an expression in which the same variable (name) is uses both locally bound and globally free.

Clear improvement

Remarkably, we are using 99% of the same apparatus as the standard approach, but there are clear improvements:

  • There is no use of limits. All information is contained in the algebra of both the function f and the dynamic quotient. See here for continuity.
  • There is a clear distinction between the three realms {x, y}, {Δx, Δy} and {dx, dy}.
  • There is the new tool of the {dx, dy} space that can be used for analysis of variations.
  • Didactically, it is better to first define the derivative in chapter 1, and then introduce the differentials in chapter 2, since the differentials aren’t needed to understand chapter 1.
  • There is clarity about the error, that one doesn’t take d≈ Δx but considers ε = Δf  s Δx, where s has been found from the recipe s = f ’[x] = {Δf // Δx, then set Δx = 0}.
Example by Van Dormolen (1970:219)

This example assumes the total differential of the function f[x, y]:

df = (∂f // ∂x) dx + (∂f // ∂y) dy

Question. Give the slope of the tangent in the point {3, 4} of the circle x² + y²  = 25.

Answer. The point is on the circle indeed. We write the equation as f[x, y] = x² + y²  = 25. The total differential gives 2x dx + 2y dy = 0. Thus dy // dx = – x // y. Evaluation at the point {3, 4} gives the slope – 3/4.  □

PM. We might develop y algebraically as a function of and then use the +√ rather than the -√. However, more abstractly, we can use [x], and use dy = g ‘[x] dx, so that the slope of the tangent is g ‘[x] at the point {3, 4}. Subsequently we use g ‘[x] = dy // dx.

PM. In the Dutch highschool programme, partial derivatives aren’t included, but when we can save time by a clear presentation, then they surely should be introduced.

Conclusion

The conclusion is that the algebraic approach to the derivative also settles the age-old question about the meaning of the differentials.

For texts in the past the interpretation of the differential is a mess. For the future, textbooks now have the option of above clarity.

Again, a discussion about didactics is an inspiration for better mathematics. Perhaps research mathematicians have abandoned this topic for ages, and it is only looked at by researchers on didactics.

Appendix A. Spiegel (1962)

Quote from Murray Spiegel (1962), “Advanced calculus (Metric edition)”, Schaum’s outline series, p58-59.

1962-spiegel-p58-59-gray

Appendix B. Adams & Essex (2013)

The following quote is from Robert A. Adams & Christopher Essex (2013), “Calculus. A Complete Course”, Pearson, p236.

  • It is a pity that they use c as a value of x rather than as an universal name for a constant (value on the y axis).
  • For them, the differential cannot be zero, while Spiegel conversely states that it is “not necessarily zero”.
  • They clearly show that you can take f ‘[x] Δin in {Δx, Δy} space, and that you then need a new symbol for the outcome, since Δy already has been defined differently. However, it is awkward to say: “For such an approximation, the quantity Δx is traditionally denoted as dx (…)”. It may well be true that John from Los Angeles is called Harry in New York, … etcetera, see above.

2013-03-adams-calculus-acompletecourse-p236-figure

2013-03-adams-calculus-acompletecourse-p236-text