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Calculus help!

bignickel

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bignickel
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HELP!

I'm going thru Martin Gardner's upddate of Thompson's "Calculus Made Easy" and there on pg 70 (hardcover) he starts doing alabraic division and I am lost.

It seems easy enough from the getgo: differentiate y = u/v. But once you add dy, du, dv things get very complicated.... And I am quite lost with what Thompson is doing.

I once stumbled across a web forum for math and science which would have allowed me to enter the equations with MathML, but I can't find it again! Googling "calculus forum" has just resulted in endless mentions of newsgroups (very helpful, that).

Anyways, let me know if you can help me, and I'll email you a word doc or a .jpg/.bmp drawing of the equations.

If you could point out what the heck he's doing and why, I'd be grateful.

Heck, if you could point me to the math forum that allows posts in mathML, I could just do that.
 
Here's a simple tip. Don't differentiate the quotient, ie. u/v.

Try differentiating the product, i.e y=u.v^-1 (v raised to the power of minus 1) and factorise.

So it is dy/dx=du/dx * v^-1 + u * d(v^-1)/dx
 
Drooper said:
Here's a simple tip. Don't differentiate the quotient, ie. u/v.

Try differentiating the product, i.e y=u.v^-1 (v raised to the power of minus 1)....
Click to expand...
This is a great tip. When I took calculus, this technique was never suggested. But I later stumbled upon it on my own, and in some cases, it saved me a heck of a lot of work.
 
Drooper said:
Here's a simple tip. Don't differentiate the quotient, ie. u/v.

Try differentiating the product, i.e y=u.v^-1 (v raised to the power of minus 1) and factorise.

So it is dy/dx=du/dx * v^-1 + u * d(v^-1)/dx
Click to expand...
...and here's an even simpler technique that avoids the terrors of differentiating v^-1:

Start with this:

y = u/v

Now multiply both sides by v:

v y = u

Now differentiate both sides (use the product rule on the left):

v' y + v y' = u' <---- (v' means dv/dx)

Use the fact that y = u/v to substitute in u/v in place of y:

v' (u/v) + v y' = u'

Now solve for y' using simple algebra:

v y' = u' - v' (u/v)
v y' = (u' v - v' u)/v
y' = (u' v - v' u)/v^2

ta-da! That's what got me through it the first time around...
 
Grey Wanderer, I've emailed it to you.

To any who wish to see the problem, PM me or let me know your email, and I can send it to you. The forum won't let me upload .doc files.
 
Huh, the message didn't mention zips, but there at the bottom, it's listed.

Just got back from lunch, so have to check email.

Here's what I copied and saved into wordpad.
 
Ok, I didn't understand the part you didn't understand. But you realise that he's just proving a simple formula for differentiating u/V? So if you don't care about the proof you can just skip to the final equation.
 
A good mnemonic for remembering the rule.

Instead of u and v, use "hi" and "ho." Then it becomes "hi d ho minus ho d hi over ho ho." Say it out loud.
 
epepke said:
A good mnemonic for remembering the rule.

Instead of u and v, use "hi" and "ho." Then it becomes "hi d ho minus ho d hi over ho ho." Say it out loud.
Click to expand...

Oh, that's very helpful.

Grey, I DO want to know what he's doing, cuz he'll just spring that stuff on me later. Gardner didn't put any notes in that area, so evidently it's obvious to him (and not to me)
 
BTW, Vorticity, I understood your proof with no problems. If Thompson had skipped whatever the heck he's doing, I wouldn't have had to bother...
 
This is how I figure it out when I forget the rule:

y = u/v

Take ln of both sides:

ln y = ln u - ln v

Differentiate:

y' / y = u' / u - v' / v where y' = dy/dx etc.

Multiply by y, substitute u/v for y and simplify.

To me, the manipulations are simpler using this method.
 
Oh....my......

I just started my first Calculus class (in highschool, an introduction before we get to university), and so far we're just doing a review of everything we've done so far in high school maths that we'll need for the rest of the year.

What's the difference between calculus and 'normal' math?
 
Originally posted by Kilted_Canuck
What's the difference between calculus and 'normal' math?
Click to expand...
Oh, there's a huge difference. Infinite, in fact.

Or maybe it's just infinitesimal, I forget . . .

:D

Yeah, limits basically.

Also, you need to get used to the idea of treating a function as a single mathematical object that you can do things to as a whole. In 'normal math', addition or multiplication or taking the square root or finding the sine, etc. are things you do to numbers to get other numbers. In calculus, differentiation and integration are things you do to functions to get other functions.

A good understanding of quantification is important. "There exists an x such that for all y ..." is very different from "for all x there exists a y such that ...". This is essentially the difference between pointwise convergence and uniform convergence, for example. (Yeah, I know, that probably means nothing to you.)
 
Originally posted by bignickel
I DO want to know what he's doing
Click to expand...
long division.

Here's how it goes, step by step (follow along in the .doc file):

On the first line, "v + dv" is the divisor, and "u + du" is the dividend. The thing on the right is the quotient, but pretend it isn't there right now, because we haven't figured it out yet.

Divide the first term of the divisor (i.e., v) into the first term of the dividend (i.e., u). Write the answer (i.e., u/v) as the first term of the quotient. Next, multiply it by the entire dividend (i.e., v + dv). This yields u + u dv / v; write that on the second line. Subtract it from the dividend (i.e., u + du); write the answer on the third line.

Now, you basically repeat the steps, using the third line as the new dividend, to get the next term of the quotient.

And then once more, for the fifth line.
 
69dodge said:

On the first line, "v + dv" is the divisor, and "u + du" is the dividend.
Next, multiply it by the entire dividend (i.e., v + dv).
Click to expand...

Thanks for your help 69dodge. I'm going thru it right now.

Got confused tho: I thought "u + du" was the dividend?
 
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