2016ers Chit-Chat Thread (13 Viewers)

[Paradoxica]

Did he get the marks for the Pascal's theorem one? Since he basically assumed the result if he just quoted the theorem as his proof.

The theorem was modified almost beyond recognition, but I think he got all the marks for it.

 

[Paradoxica]

Is it possible to construct a bijective function in which the codomain consists strictly of transcendental numbers?

 

[MilkyCat_]

Is it possible to construct a bijective function in which the codomain consists strictly of transcendental numbers?

Yes

 

[leehuan]

In fact, this can be said for most (or at least many) of the HSC maths topics, can't it?

At least with stuff such as Thales' theorem the statement of the theorem itself gets memorised. Here, they just say something like "therefore x = ..." without no justification. But yep.

Well that's the same with the intermediate value theorem and the fundamental theorem of calculus. Theres no point formalising things and naming them properly if students aren't doing rigorous analysis. (Which they cannot even begin to do without proper construction of the reals and discussion of limits, which is not at all feasible for a secondary school course).

What is important though, is that everything done in HSC mathematics CAN be made rigorous with sufficient knowledge. This is why they consult with academics when it comes to writing HSC MX2 exams.

Same principle I guess, but at least they are shown that it is indeed called the fundamental theorem of calculus. They take it granted from there.
____________________________

Obviously those questions in the HSC are not made by ordinary maths teachers. Whilst I believe any maths teacher that isn't bad can do the entire paper, not that many are going to be able to create those targetting band 6/E4 type questions.

Even my teacher once alluded to "some professor making up this question whilst he was having a cup of coffee..."

 

[leehuan]

Yes

Did you even understand much of what he said.

If so then what.

 

[MilkyCat_]

Did you even understand much of what he said.

If so then what.

Nope

 

[seanieg89]

but at least they are shown that it is indeed called the fundamental theorem of calculus.

Less often than you might think. A pretty large proportion of my ex-students had not heard that name, and just knew that "differentiation and integration were inverse to each other."

 

[leehuan]

Nope

Exactly.

Bijective function - The function is both injective and surjective
- Surjective - For every value in the codomain, there exists at least one corresponding value in the domain
- Injective - For every value in the domain, there exists at most one corresponding value in the codomain.
*Domain - The set of allowable input values of a given function

Codomain - The set of allowable output values of a given function

Transcendental numbers - A subset of all irrational numbers; the set of irrational numbers that cannot be expressed as the roots of any non-zero polynomial with rational coefficients.
-Rational numbers - The set of all numbers that can be expressed in the form p/q, where p and q are integers with the exclusion q can't equal to 0

Please tell me you knew two things in that at least

 

[leehuan]

Less often than you might think. A pretty large proportion of my ex-students had not heard that name, and just knew that "differentiation and integration were inverse to each other."

That's quite horrific. Even maths in focus mentions the name Fundamental Theorem of Calculus

 

[InteGrand]

Less often than you might think. A pretty large proportion of my ex-students had not heard that name, and just knew that "differentiation and integration were inverse to each other."

I think it is also common for HSC students to think something like d/dx ∫f(x) dx = f(x) is an application of FTC.

 

[MilkyCat_]

Exactly.

Bijective function - The function is both injective and surjective
- Surjective - For every value in the codomain, there exists at least one corresponding value in the domain
- Injective - For every value in the domain, there exists at most one corresponding value in the codomain.
*Domain - The set of allowable input values of a given function

Codomain - The set of allowable output values of a given function

Transcendental numbers - A subset of all irrational numbers; the set of irrational numbers that cannot be expressed as the roots of any non-zero polynomial with rational coefficients.
-Rational numbers - The set of all numbers that can be expressed in the form p/q, where p and q are integers with the exclusion q can't equal to 0

Please tell me you knew two things in that at least

Hahaha yeah I knew two

 

[seanieg89]

Is it possible to construct a bijective function in which the codomain consists strictly of transcendental numbers?

Yes. The function with domain {0} and codomain {e}.

For something less silly, I will assume you are asking whether or not there exists a bijection from the transcendentals to the reals. This is also true. In fact we can replace the transcendentals with any co-countable set S.

Picture the argument as follows:

Enumerate the elements of the countable complement of S as x_1,x_2,...

Enumerate an arbitrary countable subset of S as y_1,y_2,...

Now map y_{2k-1} to x_k, map y_{2k} to y_k, and leave all other elements of S fixed.

This map is then a bijection from S to R.

 

[seanieg89]

That's quite horrific. Even maths in focus mentions the name Fundamental Theorem of Calculus

Eh. I don't think it's that big a deal if students don't know what things are called in high school. I'd much rather them develop good problemsolving skills than learn the names of theorems.

FTC in particular refers to so many different theorems as we generalise notions of integration/differentiation and consider inversion in the two different orders.

 

[Nailgun]

I think it is also common for HSC students to think something like d/dx ∫f(x) dx = f(x) is an application of FTC.

Isn't the fundamental theory thingo that for a function f(x) where F(x) is the primitive function, the definite integral between b and a of f(x) is equal to F(b)-F(a)

inb4 completely wrong lel

 

[Paradoxica]

Eh. I don't think it's that big a deal if students don't know what things are called in high school. I'd much rather them develop good problemsolving skills than learn the names of theorems.

FTC in particular refers to so many different theorems as we generalise notions of integration/differentiation and consider inversion in the two different orders.

This.

Quite frankly, the majority of teachers don't care about the how and why of mathematics, only the what.

And we wonder why our mathematics is falling behind the rest of the world's...

 

[leehuan]

Isn't the fundamental theory thingo that for a function f(x) where F(x) is the primitive function, the definite integral between b and a of f(x) is equal to F(b)-F(a)

inb4 completely wrong lel

There's actually 2 statements in the fundamental theorem of calculus. That's the second.

 

[iforgotmyname]

The more you cram the less effective it becomes. Fact.

Knowledge is inversely proportional to cramming

No, thats too general. Different methods for study works with different people. Stop trying to exert your ideals on others.

 

[leehuan]

No, thats too general. Different methods for study works with different people. Stop trying to exert your ideals on others.

Except I'm not.

I have never seen a crammer beat every single person who actually studied in their cohort.

 

[iforgotmyname]

Except I'm not.

I have never seen a crammer beat every single person who actually studied in their cohort.

Lol you should meet me

 

[leehuan]

Lol you should meet me

?

You're saying you studied starting at the earliest the day before every single task you had and you got dux?