Monthly Archives: August 2014

Calculus without limits 3

So you tried, or you didn’t, now here is the derivative of 1/sqr(x)


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George, to his teacher:

I have now integrated my preconceived ideas and the enlightenments engendered by yourself, but I still have trouble differentiating between “the limit of” and “the limits of”.

George’s teacher, aside:

I think George would be better off writing a novel. he could call it “The Limits of Continuity”.

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“I did my best to pass the test”

I did the sums, no hesitation.
But then it asked for explanation.
“I know it’s right”, I wrote down fast,
“I understood from first to last!”.
“I’m going to be a mathematician,
“Not a fingernail technician!”.

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Read this : Negative numbers, by A. N. Whitehead

Alfred North Whitehead, professor of mathematics and philosophy, and famous for his collaboration with Bertrand Russell on their joint effort, the Principia Mathematica, also wrote a book, “Introduction to Mathematics”, in 1911, for High School students and others who really wanted to know what math was all about.
The section on negative numbers is so relevant to the teaching of that topic today that it is a MUST READ. Click the link to download this section.

Alfred North Whitehead: Introduction to Mathematics

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Calculus without limits 2

As h approaches zero
I quietly despair.
It really is the limit.
Please don’t take me there.

The funny thing about the calculus is that it was brought into existence by Isaac Newton in 1666 or earlier, and was developed and used without the idea of limits for over 150 years. The first attempt to get rid of the troublesome infinitesimals was by Cauchy in 1821, where he introduced the chord slope (f(x + h) – f(x))/h. The whole business of finding a satisfactory definition of the derivative was finally achieved by Weierstrass in the mid 19th century.

So here we go with cubics, and the same approach can be used for any whole number power of x, even negative ones. You should try it.


Next time  sin(x) and cos(x), so no more  sin(h)/h stuff.

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Calculus without tears (that is, without limits)

“As h approached zero I reached the limit of my understanding.”

So it seemed to me that calculus without limits would be a good idea.

Not just for powers of x, but also for trig, exp and log functions.

This is the first of several posts on this subject.

calculus1 pic


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Infinity, a place beyond.

That most strange place, infinity,
Is somewhere I don’t want to be.
I’d rather stay with Brouwer
In his ivory tower.


and for something lighter try Heavy Man


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Lament to the Common Core geometry

Could I move this trapezoid
To that one, in the endless void?
I tried translation and rotation.
Then I had a crazy notion.
I would pass a rigid motion.
Result – a lovely hemorrhoid.

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Fractions, at home and away

In the morning Johnny’s mom
Said “Here’s six candies for your break.
“Give your sister half of them”.
Now Johnny’s brain is on the make.
He gives her one, and then another.
Little sister stamps her feet!
“And the last one!” says his mother.
“Damn” thinks Johnny, “I can’t cheat!”.

Later that day

“Johnny, what is half of six?”.
“I dunno”.
“Well, go get out six lego bricks
“And make a row.
“Now break the row right in the middle.
“That’s half the row.
“Just split the half and count the bricks”.
“I got three”.
“So now you see, three’s half of six”.
But does he know?

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What is an equation? . . . . .What is NOT an equation?

Before starting, a definition: Any combination of numbers and letters and arithmetical operations (including   =   <   >   <=   >=) with more than three symbols is an algebra “thing”.  So, in passing, observe that a “number sentence” is an algebra “thing”.

Equations are neither true nor false: Some examples –

x + 10 = 45
x + 2y = 8
x^2 + y^2 = 4
y = x^2 + 5x + 7
x^2 + 5x + 7 = 0
x = 35
ax + by + c = 0

In each case the equation specifies the value or values of the letter quantities

x + 10 = 45
The value of x is such that if I add 10 to it I get 45

x + 2y = 8
The values of x and y are such that twice the y value added to the x value gives me 8

x^2 + y^2 = 4
The values of x and y are such that the square of the x value added to the square of the y value is equal to 4. This and the one above specify pairs of values.

y = x^2 + 5x + 7      You do these two
x^2 + 5x + 7 = 0

x = 35
The value of x is specified to be 35
and lastly,   ax + by + c = 0
Then we have identities, sometimes called equivalence statements.
These are ALWAYS true.

3 + 2 = 5
4 = 1 + 3
8 = 11 – 3
(x + 1)^2 = x^2 + 2x + 1
2x(4 + 7) = 2×4 + 2×7 (where x is multiplied by)
6/8 = 3/4
Later on, in algebra, we get definitions:

f(x) = 3x + 2
This means “The rule for the function whose name is f and whose input is x is multiplythevalueofhteinputxbythreeandaddtwotoit”
or “The value of the output of the function f for input x is the value of 3x + 2”.

y = f(x)
This means “The value of the output of the function f for input x is to be given the name y”.
These are NOT equations and they are NOT identities.
The whole current mess arises from the use of the equals sign for “gives” or “makes”, or “we get”, as in “3 + 5 makes 8”, or  “If we multiply 4 by 6 we get 24”, and we write

3 + 5 = 8, and 4 x 6 = 24
3 + 5 is 8 and 4 x 6 is 24 would be better.

The newfangled term “number sentence” appears to have been invented in order to avoid dealing with the correct mathematical jargon, but I see it as making things EVEN worse,


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