HSC 2016 MX2 Marathon (archive) (2 Viewers)

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seanieg89

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Re: HSC 2016 4U Marathon

Yep. What Integrand said.

All you can deduce about the geometry of the set of four roots is that this set is closed under the map z->1/z.

(In addition of course to being closed under conjugation, if our coefficients are real.)
 

seanieg89

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Re: HSC 2016 4U Marathon

A short question to test student understanding of differentiability:

Suppose the function f is differentiable to all orders at the real number a.

Explain why and as tends to zero.

What about the quantities and ? Do they tend to zero as well? Why/why not?

Recall that a function is said to be differentiable at precisely if . Could we replace the with to obtain an equivalent definition? Why/why not?
 
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agha

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Re: HSC 2016 4U Marathon

A short question to test student understanding of differentiability:

Suppose the function f is differentiable to all orders at the real number a.

Explain why and as tends to zero.[/tex]






























What about the quantities and ? Do they tend to zero as well? Why/why not?












Recall that a function is said to be differentiable at precisely if . Could we replace the with to obtain an equivalent definition? Why/why not?




 

agha

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Re: HSC 2016 4U Marathon

 
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seanieg89

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Re: HSC 2016 4U Marathon

A game of chance uses a simple harmonic oscillator. Its amplitude of oscillation is 1 but its frequency is unknown to you.

To play the game, you have to pay $1, and you get to choose a finite number of closed subintervals of [-1,1].

You win if after a random large amount of time, the oscillating particle lies in one of your subintervals, and you win $(1/L) where L is the sum of the lengths of your subintervals. So for example, if you were to cover the whole interval [-1,1] with a single subinterval, you will always "win", but as you only get $0.50 of your $1 back, this is not a good strategy!

Q/ How large an expected profit can you obtain with a well chosen bet? Provide proof.
 
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