Differential Calculus

Abstract: This paper will examine the methods of differentiation developed by the infamous pseudoscientist, Leonardo da Pantload. It will be shown that his method for finding the derivative of an arbitrary function is flat-out worthless. As it currently stands, his method will only work for a quadratic equation. For all other functions, a simple average is computed; and more often than not, this average value is far wide of the mark.

So What Exactly Is A Derivative?
Let’s begin by nailing down just what a derivative is. According to Mathis, the derivative is a procedure that flattens or straightens out a curve:

“The derivative flattens out the curve… we are straightening out the curve.” – Miles Mathis (Exponential Functions)

That’s hogwash; pure nonsense. Unless Mathis is grabbing both ends of the curve and pulling, the curve remains unchanged. Finding a derivative doesn’t have any affect on the curve whatsoever. The derivative is just the slope of the curve. To be more precise, it is the slope at any point on the curve. Derivatives are slopes; or phrased differently, rates of change.

Let’s look at an example and see exactly what the derivative is telling us. Suppose we have a function f(x) = x3. Using the power rule we can easily find its derivative, which is just f'(x) = 3x2. Now this value of 3x2 is the slope; it’s the slope at any point on the curve. For instance, if we arbitrarily pick a point on the curve, let’s say x=1, then the slope of the curve at that point is equal to 3x2 = 3(1)2 = 3. (See Figure 1)

Figure 1

Similarly, we can find the slope at another point on the curve; for instance, when x=1.5. Notice that the slope of the tangent line changes as we move along the curve from one point to the next. But the curve itself, never changes. There isn’t any flattening or straightening occurring, as Mathis has claimed. The slope of the tangent line changes from one point to the next, but everything else remains the same. (See Figure 1a)

Figure 1a

As we move along the curve, the derivative tells us the slope of the curve at each point along the path. Also notice that we can pick any point on the curve; we aren’t restricted to intervals of one, as Mathis contends:

“If you go below Δx=1, you will change your curve.” — Miles Mathis

What in heaven’s name is Mathis babbling about? The curve doesn’t change at all; the slope of the curve varies from one point to the next, but the curve itself is cast in stone, so to speak, and never changes.

The Mathis Method Of Differentiation:
Now let’s have a look at the Mathis method of differentiation. What Mathis has done is create a table of differences. The main problem with this difference table approach, is that it simply doesn’t work. The tables are limited to integers only, and don’t work at all for anything but a simple quadratic equation. For an arbitrary function, his tables are altogether worthless.

Once Mathis leaves the quadratic and begins to examine other functions (exponential, square root, log, ln, etc.), his tables fail to turn up any recognizable pattern. So what does Mathis do? He computes a simple average; that’s it. His re-definition of the derivative is just the average slope over an interval of 2:

slope @ (x,y) = [y@(x + 1) – y@(x – 1)] / 2

News Flash! Computing an average doesn’t require differentiation, or any advanced mathematics whatsoever; simple arithmetic will suffice. Also, approximations were already known, long before calculus ever arrived on the scene. The calculus was invented to provide an exact solution; everyone already knew how to compute a simple average.

Let’s take a look at an example that shows why Mathis’ simple averaging method is inadequate. Here is the basic square root function f(x) = sqrt(x). Let’s pick an arbitrary point on the curve; for instance, x=1. (See Figure 2)

Figure 2

Now, using this same square root function, let’s compare the above method to the Mathis averaging method. Mathis evaluates the function, first at x=2, and then at x=0; using the following formula. As you can see, this is just the average value over an interval of 2. (See Figure 2a)

slope @ (x,y) = [y@(x + 1) – y@(x – 1)] / 2

Figure 2a

Looking at the graph you can see that the tangent line isn’t a true tangent to the curve; it’s passing through two points. Hell, this doesn’t even qualify as a close approximation. As the above graph shows, the Mathis method of differentiation is not an exact solution; it’s just the average value. Also notice that in the region between zero and one, his equation for the slope fails entirely; it would require the square root of a negative number. For instance, when x=0.5 his slope equation evaluates as follows:

y = sqrt(x)
slope @ (x,y) = [y@(x + 1) – y@(x – 1)] / 2
slope @ (0.5, 0.707) = [sqrt(1.5) – sqrt(-0.5)] / 2
Error; invalid operation

Here is one more example: f(x) = aX. For this function Mathis does a full analysis of the problem. However, he doesn’t supply a graph showing the curve with the tangent line. In fact, as a rule, Mathis never shows the curve and the tangent line together [Red Flag]. If he ever did this, it would be obvious to everyone that his techniques are seriously flawed. Here is a summation of his analysis:

“With a=2, x=2, we found a value of 4 for the derivative and of 3 for the slope. Which is correct? Both are correct, and either can be used in math or physics.” — Miles Mathis (Exponential Functions)

Huh? Both are correct? Mathis is hopelessly confused; he can’t even decide which value to use for the slope of the tangent line. Is it 3, or is it 4? According to Mathis, you can use “either” value (you have to be monumentally stupid to say something that ridiculous). (See Figure 2b)

Figure 2b

As you can see, the tangent line is passing through two points on the curve; so it’s definitely not an exact solution. It’s only an approximation, and not a very good one at that. The correct value, shown below, is 2.77. (See Figure 2c)

Figure 2c

Before moving on to Fermat’s method of differentiation and the formal definition of a derivative, here is one more quote from Mathis. Electrical engineer’s in particular, should get a laugh from this foolhardy and naive statement:

“Finding the derivative of exponential functions is mathematically impossible, by the current methods. Exponential functions aren’t differentiable…Therefore we will never have to differentiate it in physics.” — Miles Mathis (Exponential Functions)

Hint: voltage across a capacitor.

Fermat’s Solution To The Tangent Line Problem:
To fully understand differentiation, we must first examine Fermat’s solution to the tangent line problem. Isaac Newton wrote that his own early ideas about calculus came directly from “Fermat’s method of drawing tangents”.

Unable to find the exact solution to the tangent line problem (no one has ever cracked that nut), Fermat came up with an ingenious idea; ignore the tangent line altogether and use a secant line instead. The secant, as you know, is a line that passes through two points on the curve; whereas the tangent line passes through a single point only.

And if these two points on the curve are brought sufficiently close together, the secant line and the tangent line become virtually indistinguishable (approximate equality). In fact, Leibniz defined the tangent line as the line through a pair of infinitely close points on the curve.

In a sense, Mathis has the right idea; his slope formula creates a secant line that passes through two points on the curve; that part is correct. But the interval of 2 is what ruins his solution. Were he to shrink the interval all the way down to an infinitesimal, his slope formula would be correct. But then, that would merely transform his slope equation into an exact replica of our current definition of a derivative.

The Formal Definition Of A Derivative:
Fermat made one big mistake though; he shrunk the interval (the gap between the two points on the curve) all the way down to zero. It’s a subtle, but nevertheless, vital distinction; the interval cannot equal zero. And so it was Newton and Leibniz who finally put the full puzzle together: the interval must be infinitely small, but no smaller. And that is where the concept of limit enters the picture; the interval approaches zero, but it never becomes zero. The interval shrinks to an infinitesimal.

So the derivative of a function can best be described this way: it is the slope of the secant line which passes through two infinitely close points on the curve, and is expressed mathematically as follows:

f'(a) = \lim \limits_{h \to 0} \frac{f(a + h) - f(a)}{h}

***** Work in progress… More coming soon. *****

References:
Math Before Calculus – Four Problems
Derivative for Powers (Mathis)
A Re-Definition of the Derivative (Mathis)
Derivative of Exponential Functions (Mathis)

Note: This paper was last revised on 07/30/2013

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32 thoughts on “Differential Calculus

  1. Instantaneous velocity, defined at some instant t, is the anticipated physical event (displacement or motion) that would occur over a finite, NON ZERO time interval following the instant t, provided we have uniform motion.

    To reach the above notion, one would have to recognize the following:
    An instant of a turtle in motion and an instant of a Ferrari in motion are not the same, in the sense that for the turtle, we would anticipate smaller delta x / delta t following that instant than we would for the Ferrari (hopefully if the Ferrari is working properly, if it’s not, we would be anticipating a different delta x / delta t for it, wouldn’t we Miles??).

    A turtle and a Ferrari are put to a race. The race starts at t=0. Now let’s freeze the video at t=5. What velocity would you ANTICIPATE for the turtle once the video resumes play again, provided you assume uniform motion for it? What about the Ferrari? That would be your instantaneous velocity (defined at t=5) respectively.

    Instantaneous velocity is only an anticipation or EXPECTATION (of an event). And that anticipated event occurs over an anticipated NONZERO time interval. There’s no contradiction once you recognize this.

    • A very successful Broadway producer said after establishing a successful show he turned it over to his underlings but made it a point to periodically sit in the audience to see the “improvements” and take them out.

      With this as a backdrop here is Newton’s description taken from the Principia:
      “ For by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after , but at the very instant it arrives: that is , that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, not afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be”.

      • It was that very definition in the Principia that led to the criticism leveled by Bishop Berkeley; which he cleverly referred to as “the ghosts of departed quantities”. Consequently, mathematicians then spent the next 150 years trying to provide a rigorous definition of an infinitesimal.

        Well in 1872, the whole matter was put to rest when the infinitesimal officially became a “departed quantity” itself. Mathematicians adopted the epsilon/delta definition of a limit, which doesn’t require the use of infinitesimals; thus side-stepping Berkeley’s criticism altogether.

  2. Miles, you are mixing up terminology and misrepresenting many of the basic concepts in physics. Physical events happen over time intervals, but instantaneous velocity is not a physical event. It’s a PERCEIVED or ANTICIPATED physical event. It is not defined as a physical event, it is defined as an anticipated physical event, an event that WOULD happen IF some conditions followed, which I will explain bellow:

    Instantaneous velocity does NOT comprise an event happening over a zero time interval, which of course would be a contradiction; instantaneous velocity, defined at instant t, is the ANTICIPATED delta x / delta t that WOULD occur over a HYPOTHETICAL finite time interval starting at t and ending at t+ delta t, provided that the object continues motion uniformly.

    According to physicists, instantaneous velocity is the velocity that IF, you allowed for it to act over a time interval, (while preserving it’s magnitude), it would lead to a certain displacement over that interval.

    Did you just get what I said? Most teachers don’t really understand what they teach, and most geniuses have a hard time conveying their thoughts to the average. What I just said about instantaneous velocity is what a genius like Newton would have in mind, but may not be able to explain in a convincing manner.

    Instantaneous velocity is NOT a velocity over a time interval. Instantaneous velocity is the velocity at a frozen instant. How could we possibly define velocity over zero time interval?? With no motion, no time, just a frozen picture, where everything is constant including time and place?? Let me explain.

    Instantaneous velocity is defined NOT over a time interval, but at some time t. Since “real” velocity, which physicists call “mean velocity”, like any other physical event, occurs over time intervals, they’ve had to use a different word to differentiate between a jet frozen at 1000 km/hour and a turtle frozen at 5 meters/hour.

    SO what IS instantaneous velocity?? Instantaneous velocity is a quantity that is defined at time t as follows: at time t, what is the displacement per time (delta x/ delta t) that has not yet occurred, but WOULD occur if the object continued for a finite time interval starting from t with uniform motion?

    If a car is frozen at t= 5 seconds, it’s instantaneous velocity, according to the above definition, would be its delta x/ delta t for any finite delta t starting from t= 5, provided the car preserves uniform motion in this time interval.

    It’s a thought experiment. Over some anticipated time. If the car preserves whatever instantaneous velocity it has at t=5 seconds over a finite time interval starting from t=5 seconds, say t = 5 to t= 10 seconds, what would its mean velocity (delta x/ delta t), be over this finite time interval?

    This is the exact same notion as the tangent to a curve at a point. It’s how the path would continue from that point over a finite interval provided it preserves uniform motion from that point. So instantaneous velocity at t = 5 is the mean velocity from t= 5 to t= 5 + delta t, provided we have uniform motion. THAT is instantaneous velocity at t=5, not some event paradoxically taking place at the instant t= 5.

    You don’t even need to determine the magnitude of the finite interval over which you are calculating delta x/ delta t because the ratio is constant for uniform motion (the tangent of a slope is constant because slopes are straight lines, it doesn’t matter how big an interval you choose to calculate its tangent).

    There IS a difference between an instant of a Ferrari and a turtle, and all Newton did was recognize and quantify this difference. You can recognize this difference by conducting the following thought process:
    Imagine a turtle and a Ferrari starting a race at t=0. You’ve frozen the video at t=5 seconds, no motion, no time passing, no velocity because you have no interval. Now play the rest of the video in your mind, imagining that the Ferrari and turtle would each continue their motions from t=5 seconds uniformly.

    After an arbitrary time interval has passed, like say 1 second, how much displacement have the Ferrari and turtle each made? Divide that displacement by the 1 second time interval, and voiala! You’ve reached Newton’s definition of instantaneous velocity (at t = 5 in this case). You can’t have (mean) velocity over zero time interval, but you can have something else, the above notion. You could call the first one velocity and the second one whatever name you want. Physicists call the first one mean velocity (velocity over a finite time interval), and the second one instantaneous velocity to distinguish between the two. There is no paradox, no claim of a physical event taking place over a zero time interval, And the only way to calculate this notion of instantaneous velocity would be to calculate the tangent of the curve at that instant, which can only be achieved by the mathematical process of limiting (which BTW, would yield the exact value of instantaneous velocity at t).

    Bottom line: INSTANTANEOUS VELOCITY is not an event (displacement), occurring at an instant, which would of course be a contradiction; It is a displacement that WOULD occur over a FINITE time interval IF the object would continue with uniform motion FROM the instant. It’s an ANTICIPATION OF MEAN VELOCITY, DO YOU GET IT NOW??

    At instant t, what velocity do you anticipate for the object? That would be your instantaneous velocity. Now go figure how to calculate it.

    • Excellent analysis!

      Mathis mistakenly assumes that the word “instantaneous” implies a zero time interval. But that is not the case. Instantaneous values are defined by using limits. And the epsilon-delta definition of a limit doesn’t allow the time interval to shrink all the way down to zero. The interval can be made as small as deemed necessary, but it cannot be allowed to equal zero; otherwise you would end up with division by zero, which is mathematically undefined.

      Leibniz argued that the time interval could be shrunk all the way down to an infinitesimal. But that set the philosophers a howling, since he couldn’t provide an unambiguous definition for the value of an infinitesimal. Just how big is an infinitesimal?

      Abraham Robinson has shown that infinitesimals can be used in a logically consistent manner, but his work has never really caught on with the mainstream and is seldom used.

      The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals.

      • “Mathis mistakenly assumes that the word “instantaneous” implies a zero time interval”.

        Not exactly. Mathis thinks the word “instantaneous” implies a physical event taking place over a zero time interval, or instant, and he’s right to think that this notion would be a contradiction; However, I’ve shown that Mathis has misunderstood instantaneous velocity since it does not imply any event occurring over zero time interval. Instantaneous velocity, defined at some instant t, is the anticipated physical event (displacement or motion) that would occur over a finite time interval following the instant t, provided we have uniform motion. This is the exact same notion as a tangent to a curve (at an instant of course).

        • Yes, I completely agree with you. Mathis does indeed misinterpret “instantaneous”. He thinks it means a “physical event taking place over a zero time interval”.

          And that is the crux of his misconception. Mathis doesn’t understand the word “instantaneous”; specifically, the unique way in which it is currently defined in physics and calculus.

    • An observer observing motion can label the ultimate time interval {instant} as rising or as setting but not both. Labeling it as both is defining instantaneous which means no motion.

  3. My numerical demonstration of differentiation got pushed down by a long quote from Blaise Pantload, so I’m reposting it. Interested readers are invited to try the below method for any x value on any polynomial, exponential, or trigonometric function. Then use Mathis’ method on those functions, and decide for yourself which is more useful for finding the slope at any value of x.

    Consider the graph of y = 4x^3. What is the slope at x=5? Differentiating gives us y’ = 12x^2, so through this method we predict that the slope at (or near) x=5 will be 300. Now let’s try plugging x=5 into the equation, and also a value that is very close: x=5.001:

    x=5
    y = 4 (5^3)
    y = 500
    so we have a point at (5,500)

    x=5.001
    y = 4 (5.001^3)
    y = 500.30006
    so we have another point at (5.001,500.30006)

    The slope between those points = rise/run
    = (500.30006-500)/(5.001-5)
    = .30006/.001
    = 300.06

    Repeating the procedure with a value even closer to x=5, x=5.0001, yields a result even closer to the prediction, .0300006/.0001 or 300.006. The closer the second value gets to x=5, the closer you get to slope = 300.

    Conclusion: Differentiation works to predict the slope of the graph at or very close to the value x=5. Applying this procedure to other x values and other differentiable functions will show that this conclusion applies there as well. As for Mathis’ method? Good luck with that.

    • Looks to me like Archimedes method of exhaustion. And it is irrefutable that the slope is getting closer and closer to 300. How can Mathis possibly dispute something so obvious and easily proved? The slope is 300; beyond all doubt.

      However, using the Mathis method yields the following:
      y = 4x3
      slope @ (x,y) = [y@(x + 1) – y@(x – 1)] / 2
      slope @ (5, 500) = [4(6)3 – 4(4)3] / 2
      slope @ (5, 500) = 304

      The Mathis method is simply flat out wrong. The slope is not 304, it’s 300. It would appear that the Mathis slope formula only works for a quadratic equation, and nothing else. I had assumed that it would work for all polynomials, but it turns out to be limited to quadratics only. I had given him more credit than he deserved. His slope formula will fail for every function; the quadratic being the sole exception.

      • I suppose we can look forward to a future Mathis article called “Why Non-Quadratic Functions Are Illogical” or something. After all, cubic and higher functions generally aren’t taught in high school. Therefore, they must be useless/illogical/a conspiracy.

      • Let us look to the slope…

        Let f(x) be a function we than have a function s(x) defined as [ f(x+1)−f(x−1) ] / 2

        Using f(x±1) = f(x) ± f'(x) +f”(x)/2 ± f”'(x)/6 + f””(x) + … we get

        s(x) = f'(x) + f”'(x)/6 + …

        It is therefore exact only for a parobola…

        When we use the f(x) = 4x³ we get f”'(x) = 24…

        So s(x) = 12x² + 4

        That explains why the real slope at x=5 is 300 while Miles gets 304…

  4. “That oldest mistake is one that Euclid made. It concerns the definition of the point. Entire library shelves have been filled commenting on Euclid’s definitions, but neither he nor anyone since has appeared to notice the gaping hole in that definition. Euclid declined to inform us whether his point was a real point or a diagrammed point. Most will say that it is a geometric point, and that a geometric point is either both real and diagrammed or it is neither. But all the arguments in that line have been philosophical misdirection. The problem that has to be solved mathematically concerns the dimensions created by the definition. That is, Euclid’s hole is not a philosophical or metaphysical one, it is a mechanical and mathematical one. Geometry is mathematics, and mathematics concerns numbers. So the operational question is, can you assign a number to a point, and if you do, what mathematical outcome must there be to that assignment? I have exhaustively shown that you cannot assign a counting number to a real point. A real point is dimensionless; it therefore has no extension in any direction. You can apply an ordinal number to it, but you cannot assign a cardinal number to it. Since mathematics and physics concern cardinal or counting numbers, the point cannot enter their equations.
    This is of fundamental contemporary importance, since it means that the point cannot enter calculus equations. It also cannot exit calculus equations. Meaning that you cannot find points as the solutions to any differential or integral problems. There is simply no such thing as a solution at an instant or a point, including a solution that claims to be a velocity, a time, a distance, or an acceleration. Whenever mathematics is applied to physics, the point is not a possible solution or a possible question or axiom. It is not part of the math.
    Now, it is true that diagrammed points may be used in mathematics and physics. You can easily assign a number to a diagrammed point. Descartes gave us a very useful graph to use when diagramming them. But these diagrammed points are not physical points and cannot stand for physical points. A physical point has no dimensions, by definition. A diagrammed point must have at least one dimension. In a Cartesian graph, a diagrammed point has two dimensions: it has an x-dimension and a y-dimension. What people have not remembered is that if you enter a series of equations with a certain number of dimensions, you must exit that series of equations with the same number of dimensions. If you assign a variable to a parameter, then that variable must have at least one dimension. It must have at least one dimension because you intend to assign a number to it. That is what a variable is—a potential number. This means that all your variables and all your solutions must have at least one dimension at all times. If they didn’t, you couldn’t assign numbers to them.

    This critical finding of mine has thousands of implications in physics, but I will mention only a couple. It has huge implications in Quantum Electro-Dynamics, since the entire problem of renormalization is caused by this hole in Euclid’s definition. Because neither Descartes nor Newton nor Schrodinger nor Feynman saw this hole for what it was, QED has inherited the entire false foundation of the calculus. Many of the problems of QED, including all the problems of renormalization, come about from infinities and zeroes appearing in equations in strange ways. All these problems are caused by mis-defining variables. The variables in QED start acting strangely when they have one or more dimensions, but the scientists mistakenly assign them zero dimensions. In short, the scientists and mathematicians have insisted on inserting physical points into their equations, and these equations are rebelling. Mathematical equations of all kinds cannot absorb physical points. They can express intervals only. The calculus is at root a differential calculus, and zero is not a differential. The reason for all of this is not mystical or esoteric; it is simply the one I have stated above—you cannot assign a number to a point. It is logical and definitional.”

    — Miles Mathis (http://milesmathis.com/central.html)

  5. Wait! I understand it now: if I presume to take an arbitrarily small interval (and call that infinitesimal) and pretend that is out of sight for logic, I can get any answer I like! I can even make people believe you can get around a corner with a straight line….isn’t that brilliant?

    • If the concept of infinitesimal is too vague, then use an actual number. Provided the number is sufficiently small, it won’t make a whole lot of difference. Instead of infinitesimal, substitute the number 1/(googol).

      Unless you carry your calculations out past 100 significant digits, you’ll still get the same result.

      • Steven O. cracks me up. Who has the time to plug in real numbers, actually try a calculation, and see what you get, when you can just dismiss the whole calculus endeavor outright as illogical because your hero said so!

        • Could you please explain the logic that if you plug in a number in formula, you prove a theory is correct? Especially if it is fitted to the data like most physics formula’s are today? Looks circular to me.

          • We aren’t even talking about physics here, or physical models. This thread is about simple differentiation, a basic mathematical procedure. And Steve’s point is correct. Consider the graph of y = 4x^3. What is the slope at x=5? Differentiating gives us y’ = 12x^2, so through this method we predict that the slope at (or near) x=5 will be 300. Now let’s try plugging x=5 into the equation, and also a value that is very close: x=5.001:

            x=5
            y = 4 (5^3)
            y = 500
            so we have a point at (5,500)

            x=5.001
            y = 4 (5.001^3)
            y = 500.30006
            so we have a point at (5.001,500.30006)

            slope between those points = rise/run
            = (500.30006-500)/(5.001-5)
            = .30006/.001
            = 300.06

            Repeating the procedure with a value even closer to x=5, x=5.0001, yields a result even closer to the prediction, .0300006/.0001 or 300.006. The closer the second value gets to x=5, the closer you get to slope = 300. Quite simple really.

            Conclusion: The process of differentiation works to predict the slope of the graph at or very close to the value x=5. There’s no “fitting” going on here, or liberties of logic taken. Just plain math. One might as well argue that multiplication does not work or is illogical.

            • Sure, whenever it suits your argument, this blog is suddenly no longer about physics…

              The calculation of the differential (function) you show remains also valid without going to smaller intervals as Miles shows in his papers, it only depends on the “rise/run” of the number line. So, what you describe provides no proof, it is just circular argument as I stated before. Taken to the limit, it is based on the assumption that an interval can be assigned to a point, which even in plain math is illogical.

              • It never was about physics, it’s about Miles Mathis being a crackpot who thinks he’s figured out that everything is wrong, and no one has figured out any of these things he has, and that no one accepts his genius ideas. Unfortunately, his work relies on NOT knowing math. ”Special Relativity requires tensor calculus? Well, I’ll just CORRECT it by not learning tensors, and if people don’t accept my marvelousness they’re just being poopyheads.”

            • Only in crank-world would a clear, stepwise numerical demonstration of a mathematical procedure’s usefulness be dismissed as illogical or circular. You should email your bank to complain about the way they’re calculating compound interest. Maybe quoting one of Mathis’ blusterfests will change their mind!

  6. I had to wade thru over a hundred pages of Miles double talk only to discover that he is just computing the average slope. Is he serious? That isn’t calculus. He has redefined the derivative to be the average slope of the curve. What a joke.

    • Yep, you hit the nail on the head. That is exactly what Mathis has done. He wrote 104 pages of worthless tripe, that ultimately culminates in a redefinition of the derivative as the average slope over an interval of two. That’s it in a nutshell. To be sure, it’s a joke; but it’s also an elaborate hoax; a brazen and unabashed farce. The average value has been known for millennia.

      Mathis has deceived his readers into believing that differential calculus is nothing more than elementary arithmetic; computing an average. The following quote reveals the full extent of his delusions; or outright lies, whichever the case may be:

      “My method is vastly superior to the current method in both operation and answer.” — Miles Mathis (Derivative Of The Natural Log)

      A simple average is vastly superior? Yeah, and Mathis is “vastly” full of crap.

      • Steve I just wanted to commend you for the fine job your doing on handing Mathis his ass. He’s been asking for it for quite some time. I also enjoy other posters contributions as well, please keep up the good work and thank you all for the public service. I don’t think you’ll be able to convince the true believers of anything, as they’re so far down the rabbit hole along with Mathis they can’t (refuse to?) even recognize the light of day anymore. For those who have doubts about the accuracy of Mathis’ claims and haven’t yet fell into his hole or fell too far, you’ve come to a good place. I suggest paying close attention to what’s being said here, there’s good reason to doubt what Mathis says.

        • Thank you. It’s reassuring to know that at least a few of you guys can see through Mathis, and have concluded that he’s a fake and a fraud — yourself, Norris, Bos, Crackpotwatch, physics freak et al.

          Unfortunately, there are a handful of deluded individuals who will believe anything Mathis tells them. If he were to announce tomorrow that pi equals 892, they would all just nod in agreement and claim that it makes perfect sense. It’s impossible to get through to them; they’ve gone down the rabbit hole, as you say.

  7. When Mathis says that a derivative straightens out the curve, I think he means that the derivative of a curved function is a straight function. Which of course is bollocks. He’s appealing to an audience that maxes out at 10th-grade algebra, who might recall that differentiating some curves, like y= x^2, yields straight lines. This is standard conspiracy-theorist technique: offer just one example the reader is familiar with, and appeal to their intuition with that example. If they’re sufficiently gullible, they’ll swallow it whole and won’t ask any pertinent questions.

    When Mathis says exponential functions are not differentiable “by the current methods,” does he mean that the derivative of y = e^x is the same curve, y = e^x? Again, mind-boggling. By that logic, it’s mathematically impossible to multiply zero by anything, because all you get is zero again. I don’t know what drives Mathis to produce this counter-intellectual trash. Web kitty money, I guess.

    BTW I laughed for about a minute at “Leonardo da Pantload.”

    • “Exponential functions aren’t differentiable… If they were differentiable, then the differentials would flatten. Not only do the differentials not flatten, or move toward a straight line, they don’t change at all.” – Miles Mathis

      By the same logic, the trig functions wouldn’t be differentiable either. Mathis seems to think that all functions are required to behave just like polynomials; that is, if you repeatedly take the derivative eventually you’ll end up with a value of zero.

    • Oh I get it. Differentiation is supposed to make a function flatten out, because Mathis once saw that happen decades ago in a high-school math class. The same way the Twin Towers must have been zapped by holograms, because he saw that happen in a movie once. It all makes sense now.

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