Recreational Problems (SM level)

[Neobeo]

1) if

is a quadratic function and its maximum value occurs at a point . is a point of

intersection of with x-axis and point is such that chord subtends a right

angle at . Find the area enclosed by and chord .

It's probably but I don't really feel like doing the tedious working out here.

[Tea.bag]

correct.
ill post more question when i can be bothered

[unknown id]

how do u prove that ?

[ed_saifa]

how do u prove that ?

sandwich theorem

[Matt The Rat]

l'Hôpital's rule would also work.

l'Hôpital's rule

[dcc]

here are some fun questions to play around with:

1.


2.


3.


4.

[Collin Li]

1.
Sandwich rule:

Since :

, for large positive .

So, since




Use the same idea with 2. to get:

[Collin Li]

By intuition, 3. is 0. Approaching from the left side will not be a problem, because you can still take the square root of a small negative number plus two.

Since :

The argument of the sine function will approach infinite as the approaches zero, which means a sandwich rule is required to deal with it.



Limits is basically using intuition first, algebra second.


For 4., just use the definition of the limit. It exists if the limits from both sides agree. Then, the limit takes the value of the limits from both sides.

and

[Ahmad]

l'Hôpital's rule would also work.

l'Hôpital's rule

That's correct. However to compute the derivative of would require the knowledge of this limit, so it would be circular to use it to prove this limit. Unless you defined some other way, i.e. maclaurin series.

[Collin Li]

Yes, in my Semester 2 notes, I remember this exact limit.

"But how do we know this limit? Notice that L'Hopital is illegal here (why? The reason is that calculating the derivative of sine uses the very limit we're attempting to determine). And, high school proofs use the geometry of sine, but what we would like is a purely analytic proof (no pictures!). This isolates the real issue: analytically, we don't know what sine is! We'll leave the discussion here: we simply want to make the point that some limits (and functions) which we usually take for granted are not nearly as simple as presented. (The same remark can be made for the function , as it is not obvious how to define . It is standard to define as the limit of . But if we do that, then it's not at all obvious why . In fact, it's not even obvious what means, say for irrational." - Marty Ross

This guy was my tutor for Applied Maths. Can you tell I was blessed with a teacher who could instil a great curiosity for mathematics?

[unknown id]

thanks guys!

[Matt The Rat]

No worries. Keep in mind what coblin and Ahmad pointed out though.

[gfb]

Hey guys,

I've got another question..
Any help will be appreciated..

( btw ignore the imperial units- the in is metres (m) and the lb is kN )



EDIT:         : I've done the question.

[ed_saifa]

Here's one to test:
The area of the union of several circles equals . Prove that it is possible to choose several of them, that do not intersect each other and whose total area is greater than

[gfb]

Determine whether the given function is a solution of the given initial value problem: