h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Calculus - A12860912
Author: Baryonic Being - save GuideML out of a word-processor: A7720562 - U234603

I wanted to write an Entry purely about mathematics, so I thought - what would be a nice simple place to start?

I don't know what happened to *that* thought, because I ended up writing about calculus.

I've seen the other attempts at writing a calculus article (such as the table I have used from A1092449) and I will leave a message on the PS of the user who was previously trying to update a Flea Market Entry about calculus.

It's important that if you can't follow this Entry - especially near the beginning - that you say where your understanding is lacking, because I really want this to be as accessible as possible.

I used my OpenOffice plugin to write this Entry (see A7720562), but a lot of things had to be amended manually!

Looks interesting.

However, I have difficulty understanding it; but I'm sure that is *by no means* a fault of your own - I just don't understand math, no matter what. You could explain it to me like I was a five year old, using coins or something and I wouldn't get it.

But this entry grabbed my interest because my sister's boyfriend, who is a math teacher, is into calculus and is writing, get this, an educational *musical* about it to take on the road later this year to perform. My sister is in the show also, doing the music and things.

http://www.ftmax.com/matheatre/index.php

If you can explain calculus to me then you, Sir, are a god among men

I'll have to read the entry another time though - got to go to work soon.

I think I shall have to go through it again, then, and see if I can thoroughly reduce it to a perfectly explicable state.

And that musical? That has *made my day*! That's utterly brilliant! I love it! You know, they should hear this song - http://www.opsound.org/artist/markengelberg/ - a biography of Ramanujan (one of the greatest mathematicians ever) put to music! We need more people in the world who aren't afraid to sing the praises of mathematics - literally!

Anyway, at what point do you get confused, 42donpanic?

Remember - understanding mathematics is *not* genetic. Anybody *can* understand it - it just has to be explained to them in the right way. They might need a lot of background, of course; it'll vary as to how much they need.

Traveller in Time not really a mathematician
"Though I would call it _a_ mathematical analysis

>Though Newton suspected Leibniz of stealing modern historians have reached the consensus to the contrary.

Can be the truth, however both only have the idea, neither can/ could prove anything or actually claim one or the other ? Then what is the statement here ?

still reading . . . "

Traveller in Time with lots of imagination
"
>zooming over a hundred kilometres in the next ten minutes
What does this distance in the story ?


-------------------------------------------------

A bit confusing: first a story about a speeding fine, (stupid as when you did exceed the limit you would still be racing above that speed)

Then something mathematical about the linear part of the graph.

Without warning or completion of the flight flowing into other functions (and graphs).


Continuing, (with flushed graph buffers )"

Hi BB,

I like this

What follows are some suggestions - ignore as you see fit




<< Though Newton suspected Leibniz of stealing his work,
<< modern historians have reached a consensus to the contrary.

Did Newton _accuse_ Leibniz of theft/plagiarism/whatever?

----

In the following paragraph, you mention (twice) 'THE calculus' - everywhere else you say 'calculus' (without a 'the')

----

<< it allows us to solve difficult problems with similar logic
<< to that of game-play

Maybe an explanatory link or footnote from 'game-play' would be helpful

----

<< To understand this, we shall use a 'real world' example, but when you...

Maybe the following might be more realistic/less optimistic

>> To explain this, we shall use a 'real world' example, but when you...

----

<< If we were to plot a graph of your movement using this information,
<< it would be a distance-time graph, in which we put time along the
<< bottom and distance up the side

>> Using this information, we could represent your movement on a
>> distance-time graph, in which we put time along the
>> bottom and distance up the side

----

<< Anyway, you're fined for speeding, but you're convinced you're in the clear

I like this However, soon afterwards, you begin mixing the words 'velocity' and 'speed'

I think you should use only one word throughout - even later on with:
>> velocity-time graph<<
AND
>> What is velocity divided by time? It's acceleration6.<<
(great footnote )

I have a hunch that the word 'speed' works well enough, whilst the word 'velocity' could/would needlessly confuse matters... or maybe a footnote somewhere saying "speed and velocity aren't the same... but... etc etc"

----

I'm not sure what the in-house rules are re fonts... but I reckon a 'geeky' font (maybe 'Courier'???) might be a good idea to highlight formulae / variables / sin / cos / tan / jargon / etc

----

At one point, it says

<< ... divided by the horizontal: 36 / 2 = 18. The proper notation is this:
<<
<< DELTA y / DELTA x = 18

later on, it says

<< ... In other words, just shun the x. The proper notation is this:
<<
<< dy / dx = 0.5

I reckon you should use DELTA everywhere, which would allow you you to use 'd' for distance (instead of 's') in >> If you wanted to, you could call the time t instead of x and the distance could be called s instead of y. <<

----

Going back a bit...

<< This is where the game begins, because by DIFFERENTIATING this function,
<< you can obtain another function that...

You haven't explained what the DIFFERENTIATE means yet... so I can see many readers getting lost at this point

----

<< If you have something like 4x^3, then you do the same thing: bring
<< the little number to the front and reduce the power by one, ...

I like this

----

<< If you have 5x^6, then, the DERIVATIVE is 6 times 5 times x^6-1, ...

This looks like the first use of the word 'derivative' - if so, I think an explanatory piece is needed somewhere - preferably as simple as earlier where you use 'the bottom' for the x-axis and "When the x has a power (or index number, which is the little number at the top)", both of which I think are great

----

<< So what can we do with this?

I really like this casual style Maybe a subheader could go somewhere here????

----

<< Well, we now know that at the maximum point on our graph,
<< our speed was equal 2 times the x value.

>> Well, we know that at ALL points on our graph (including the maximum),
>> our speed was equal to two multiplied by the time value (or speed = 2x).

----

<< The x value is ten, because it was ten minutes after we started
<< accelerating, so the gradient at this point is equal to 2 times 10 = 20.

>> Ten minutes after we started accelerating, the x value is ten. At this
>> moment, the gradient (your speed) is found by multipliying two by ten (y = 2x),
>> which equates to twenty kilometeres per minute.

----

<< x still represents the time

>> The letter x still represents the time

----

<< To differentiate 2x, we just get rid of the x, leaving us with 2.

Why not show the working? e.g.
2x is the same as 2x^1...
2x^1 differenciated = 1*2x^1-1 = 2x^0 = 2 *1 = 2

----

<< So, integrating a function of velocity gives us the distance travelled.

I know what you mean by "a function of velocity" but I wonder if beginners will... after all 'function' means something rather different in 'English' (cf maths-speak)

If that makes sense... how about something like the following instead?

>> So, integrating a formula that plots/shows/huh speed (?not? velocity)
>> will give us a formula that shows the distance travelled.

----

<< you draw the graph of y = 2x - that's the velocity function - it's just a...

those hyphens could be confusing...

>> you draw the graph of y = 2x (that's the 'speed' function), it's just a...

----

<< By putting on the upper limit at ten minutes, we can form a triangle.

>> By setting 'ten minutes' as the upper limit, we can form a triangle.

----

<< Suppose we want the area under the curve y = 9x2-2x3+5x

>> Suppose we want to measure the area under a curve defined by y = 9x2-2x3+5x

----
>> ... we have to torture the number 5 to the tune of this function.
>>
>> 3 * 5^3 - 0.5 * 5^4 + 2.5 * 5^2

Of course, order of operations means no brackets are necessary... but including them might aid comprehension

<< [(3 * 5^3) - (0.5 * 5^4) + (2.5 * 5^2)] >>

----

Summary

Hmmm... I'd expect a summary to come at the end... how about 'Recap' or something simliar?

----

I find it rather confusing to read the displacement/velocity/acceleration table... maybe adding <> might help ????

----

I think that the 'Advanced Play' section is a bit too advanced - insofar as it is v light on 'English' and heavy on 'maths notation' - which ain't helped by some rather sparse 'workings'...

I find reading
<< dy / dx = 5x^3(63x^6) - 9x^7(15x^2) / 25x^6 = 315x^9 - 135x^9 / 25x^6 = 180x^9 / 25x^6 = 7.2x^3 >>
on ONE line too confuzzling to read/check

It seems fairly obvious that you have a good grasp of both calculus AND GuideML - I'll be checking out your GuideDog for OpenOffice v soon... not that I write much for the guide

Anyhoo... how about 'fleshing out' AND formatting this last section into tables ?????

----

Cheers
six7s

Traveller in Time from another angle
"Work out the examples, like rocket flight, complete it feels like we are still searching for the answer of the first in the next paragraph.

Try to give the mathematics the same look in the series of examples. "

It's difficult to explain calculus with just text, but this is a pretty damn good job. My Maths teacher taught me with extensive use of those interactive graphing computer programs, which makes the process quite a bit easier. This is definitely a case for changing the rules to allow more freedom with pictorial content in the guide.

For the turning points section, you don't mention what happens when the second derivative is 0. It would be worth saying that it could be a point of inflection, and how to find out.

Thanks a million, six7s - that was indispensably valuable (not to mention a great help!)

I have merged most of your changes (you're familiar with the Unix 'diff' program, perhaps?). Your suggestion of using DELTA throughout can't be done, unfortunately, because the notation for differentiation uses 'd'. The notation that uses DELTA is not differentiation. Also, 's' is the standard symbol for distance in this subject.

And thank you, laconian; I have added a saddle point (a point about saddles).

I have tried to make the 'advanced play' section more comprehensible, but I've ended up using an analogy with the Flinstones - maybe I've dumbed it down *too* much? See what you think.

I've mostly used 'speed' but I don't think a v-t graph is ever called a speed-time graph, so I've added an explanatory footnote for that.

I put the freestanding maths lines in Courier font, which does help to differentiate them from the prose. Thanks for that too.

And, Traveller in Time, I'm not sure what you're trying to suggest. I did say that the rocket comes to a stop, but I don't want to invent a uniform deceleration or something, because then I'd need to go through another example (probably needlessly).

I also didn't know what you meant about a footnote about game-play, six7s?

H'OK. Thanks again. Time for a break.

This is just about on the threshold of my mathematical understanding (A-Level scarped pass), but it's brilliantly done. Any more changes in the pipeline, BB?

B

Hello, Mu Beta. Nope, no changes in the pipeline unless suggestions are fed into the funnel!

http://www.bluemoon.net/~watson/proof.htm

Note 'Proof by calculus' .

At the end of the integration section you show how to find the area under a curve from 0 to something. Is it worth putting a note in that if you are looking for the area from 2 to something then you take f(2) away from f(something). And what about the area from, say -2 to 3? I suppose it depends how comprehensive you want to be. You don't want to scare people off with too much info after all!

I think the Entry as it stands cuts a good line for non-scientists. With Entries of this type, of course, there'll always be people on both sides of the divide who think it's ridiculously incomplete.

B

I hated calculus. Nobody ever explained to me why we did it (it was a 'pure' maths class) just how to do it. And I almost never got them right. However, if I'd had the explanation at the beginning of this, it would have made sense.

Right up until you do all those twiddly smoke 'n' mirror maths things (like: In this case, we get y = 3x-1, which yields -3x-2. This isn't pretty, so you should put -3/x2.)

But all the same... great entry. Well done.

Although you define what calculus is, I think somewere in the introduction you should highlight the type of problems that can be solved using calculus.

You might get some ideas from here:

http://www.bath.ac.uk/~spw20/application.html

It's worth saying that the word 'calculus' comes from the Latin for a pebble, reflecting the technique used by the Romans for reckoning (= calculating).

The word 'calculus' is also used in Anatomy/ Medicine for a stone in the body. You may be ableto make a link here; also possibly a link to one of the 'Romans' Entries.

Thank you very much! The trouble with the uses of calculus is that has so many! It's everywhere! I've linked to that page, and summarised its contents.

I also added an etymology section, though I think I got carried away with the links. I never I thought I'd be linking to the Entries on kidneys, toothpaste and pebbles in this article! Which Roman Entry would be best, if any?

'The trouble with the uses of calculus is that has so many!

That's why I said to highlight the TYPES of problems that calculus is needed for, and WHY.
(I looked in my 'Calculus Made Easy' book and this doesn't do this either - so I think it would be a useful thing to do if you can )

Yes, I've tried all right. But it can basically analyse anything that changes... and in fact, everything in the universe changes. It's a generic mathematical technique. So, anyway, I have indeed specified the subject areas that it covers, as well as the three main things on that website. Thanks again!