Leaving Cert maths proofs

Carsinian Thau

For the LC our maths teacher gave us a list of proofs to learn about a week ago. I've kinda just started and already I've run into difficulty. Most of them I can get out by intuition but I can't get the proof of derivative formulae for products and quotients. So I had to look it up. My book uses the first principles method and it looks really complicated, but I found another method on the net called the "delta x" method. It looks to be a lot simpler.
Is it acceptable for use in the exams?

Also, one of the required proofs is for transformation geometry that says:
Each transformation f of the plane M which has coordinate form (x,y) -> (x',y'), where x'=ax+by and y'=cx+dy, and ad-bc≠0 etc.
What does this mean?!

Son Goku

Carsinian Thau wrote:

but I found another method on the net called the "delta x" method. It looks to be a lot simpler.
Is it acceptable for use in the exams?

I don't think so, I could be wrong though. The funny thing is that it is easier because it is actually correct and the first principles method is not. First principles is actually hand wavy arguement that doesn't really make any sense when you look closely at it (you can't properly prove the chain rule and other things with it). The delta x way is the correct method created by Cauchy in the 19th century. Can I see the webpage, I'll ask somebody who knows the course better than me if it is allowed.

Also, one of the required proofs is for transformation geometry that says:
Each transformation f of the plane M which has coordinate form (x,y) -> (x',y'), where x'=ax+by and y'=cx+dy, and ad-bc≠0 etc.
What does this mean?!

It means M takes the point (x,y) to a new point (x',y'). x'=ax+by and y'=cx+dy tell you how to get (x',y') from (x,y) and ad-bc≠0 is a constraint on these equations.
For instance if a=3, b=4, c=2, d=1, then the point (4,5) becomes (32,13).

Carsinian Thau

Link: http://www.leavingcertsolutions.com/mall/leavingcertsolutions/pdf/differential_calculus_maths_higher.pdf

So the statement is the proof?

MathsManiac

The delta x method and the h method are BOTH first principles and both are perfectly acceptable at leaving cert. Neither is more or less logically sound than the other, and there's nothing that can be proved with one that can't be proved with the other.

They are simply different notations for the same thing:
delta x is the same thing as h, and delta y is defined to be the quantity f(x + delta x) - f(x). [That is, f(x+h)-f(x)].

So, just as the derivative in one notation is limit[as h->0] of [f(x+h)-f(x)]/h, so in the other notation it is limit[as delta x -> 0] of [delta y / delta x].

The fact that the delta notation uses one term where the h notation uses two makes certain algebraic manipulations a bit easier, including those involved in the product and quotient rules.

Use whichever one you like, but personally I would recommend at this time of the year that you stick to what's familiar.

MathsManiac

...And on your other question, I would suggest that if you haven't done transformation geometry, it's a bit late to start now - perhaps stick to the other questions. (If you have done transformation geometry, I would have thought you would recognise what that theorem is saying.)

If you have done the topic and all you need is a reminder, here's the gist:
Transformations are functions that send points in the plane to points in the plane: for every input point you stick into the function, it spits a point back out at you. So, if you throw a whole lot of points in (like for instance all the points on a particular line) it'll spit a whole lotta points back out. The points it spits back out might or might not be a line, depending on what the function did with them; it could be a curve, or a scattering of points all over the place, or even a single point.

Anyway, this theorem tells us that a particular bunch of transformations behaves in a special way: unlike many other transformations, this particular bunch always gives you back a line when you give it a line; furthermore, if you give one of these guys two parallel lines, it'll spit back a pair of parallel lines; if you give it a line segment , it'll spit back a line segment, and if you give it a parallelogram, it'll spit back a parallelogram (although not necessarily one the same shape or size, by the way).

And the bunch of transformations that behaves in this remarkable way are the ones that are of the form given in the theorem statement. That is, where to get the coordinates of the output points, all you're allowed do is multiply the input coordinates by some numbers and add them. That is, you're not allowed go squaring them or anything mad like that. (Also, the numbers you multiply by aren't allowed to be ruled out by the little bit stuck on the end that says ad-bc not 0.)

Now, if you get what it's all about, maybe you can follow the proofs in your book or notes. (There are different methods, so I won't go into any particular one.)

Son Goku

MathsManiac wrote:

The delta x method and the h method are BOTH first principles and both are perfectly acceptable at leaving cert. Neither is more or less logically sound than the other, and there's nothing that can be proved with one that can't be proved with the other.

I've got to stop presuming I know what the OP is talking about. Sorry OP I thought you were talking about the epsilon-delta method, which people sometimes call the delta method or epsilon method. Listen to MathsManiac.

Lucas10101

The Method used via Delta H and Differentiating From First Principles are the most difficult and awkward ways to do it. The EASIEST way by far is by using the Laws of Logarithms, exceptionally easy.

MathsManiac

Lucas10101 wrote:

The Method used via Delta H and Differentiating From First Principles are the most difficult and awkward ways to do it. The EASIEST way by far is by using the Laws of Logarithms, exceptionally easy.

...but deeply unsatisfactory from a mathematical perspective, since you're using properties of a function that you haven't established on a proper basis beforehand.

COCK

i just did probability there before christmas and it struck me as probably the easiest parts on the course and something i could fall back on if the trig questions or the circle or vectors or something on paper two comes out with some ridiculous part c.

But i was just wondering that if, with your knowledge of probability provided by the textbooks could you handle the further probability question. I dont trust the further calculus question (a la integration of tan or something equally difficult) so i would like something to fall back on...any inputs?

Michael Collins

Generally no. That's why it's in a seperate section. Although having said that, from what I remember the further probability section takes only a very small amount of effort after doing the standard probability part. So if you like probablity (some don't, particularly because you can't tell if your answer is right or not), then I'd say go for it, but you will have to look through that section. Also doesn't that section include statistics? So you'll need to learn that as well...