Roots with conditions (1 Viewer)

[cutemouse]

Yes indeed. Products to sums and sums to products is VERY useful and greatly simplifies many things. And this is why it should still be taught.

Do you have an example where it could simplify things? (in the current course)

The fundamental limit is

(provided x is in radians)

Ahh okay... That explains why it is in the Coroneos 2U book, and not in the 3U book.

 

[buchanan]

Compare Fitzpatrick's 3 unit lovely proof (page 82) of the derivative of sin x to Jones & Couchman's horrible one (Book 1, page 440).

Fitzpatrick uses a sum to product and gets it out in just a few lines.

Jones & Couchman's is a page of ugly algebra nobody really wants to read.

Another nice use is to prove the tangent rule.

 

[cutemouse]

Ahh yes first principles for trig functions... I think Coroneos also uses that method. I find that his examples are more systematic than other books, which I like.

What's the tangent rule?

 

[buchanan]

Suppose you have with sides opposite angles , , named , , . Then

 

[cutemouse]

Nice.. I don't think I did that though..

 

[buchanan]

There is a proof using sums to products at http://en.wikipedia.org/wiki/Law_of_tangents

 

[buchanan]

I know 4U changed alot

Yes. To make comparisons to the old 4 unit course easier (which ran from 1966-1980), I've attached a pdf of the contents section of Coroneos' Level 1 books.

I've also attached a pdf of the contents section of an even older book by Coroneos on the Leaving Certificate Mathematics I Honours course (which incidentally was his first book) which ran from 1916-1966. This was supposed to start in 1913, but Carslaw was overruled by the University of Sydney Senate. The university had its own entrance exams (which started in 1858) which they were trying to protect. He eventually won three years later. It's just as well that he did win because the LC exams were of a higher standard than Sydney University's entrance exams.

I have attached the 1916 exam.

Level 1 (later called 4 unit) exams are at

http://4unitmaths.com/hsc1967-1974.pdf

http://4unitmaths.com/hsc1975-1980.pdf

 

[cutemouse]

Hey Buchnan, you posted the list of topics before here: http://community.boredofstudies.org/238/extracurricular-topics/102519/arc-length.html#post2220688