HSC 2016 MX2 Marathon (archive) (2 Viewers)

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Paradoxica

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Re: HSC 2016 4U Marathon

Using the Integral test, prove the following limit:



You may assume uniform convergence and the squeeze theorem.
 
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hedgehog_7

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Re: HSC 2016 4U Marathon

uhh the provided answer is 1/2 (pi(pi root 2 -1))
 

Paradoxica

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Re: HSC 2016 4U Marathon

wait how did you end up with this integral? the area i chose was just between 0 and 1/root 2, why does it include between 1/root 2 and 1?
the area bounded by the two curves are bounded by two different curves.

so you have to consider separate cases.
 

Paradoxica

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Re: HSC 2016 4U Marathon

uhh the provided answer is 1/2 (pi(pi root 2 -1))
According to Mathematica, the exact value of the integral is



A numerical approximation using Simpson's Rule confirms the result.

I guess this integral is just tedious bashing with IBP.
 

math man

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Re: HSC 2016 4U Marathon

Appliedville kicked me out, I'm just strolling these streets to find a town that will accept me
 

hedgehog_7

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Re: HSC 2016 4U Marathon

A particle is thrown vertically upwards where the air resistance is given by R = 0.01mv^2. If the velocity of projection is 60m/s find the time to reach the highest point. Answer is 3.43 s
 

hedgehog_7

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Re: HSC 2016 4U Marathon

a projectile is fired vertically upwards from the earths surface with velocity U m/s. The retardation due to gravity is given by the law k/x^2 where x is the distance of the projectile from the centre of the earth, and k is a constant. The acceleration due to gravity on the earths surface is g. The earths radius is R. Neglecting the air resistance show that if U^2 = gR, then the projectile reaches the height R above the earths surface. What is the time for this journey?


having trouble with getting the U^2 = gR bit
 

InteGrand

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Re: HSC 2016 4U Marathon

a projectile is fired vertically upwards from the earths surface with velocity U m/s. The retardation due to gravity is given by the law k/x^2 where x is the distance of the projectile from the centre of the earth, and k is a constant. The acceleration due to gravity on the earths surface is g. The earths radius is R. Neglecting the air resistance show that if U^2 = gR, then the projectile reaches the height R above the earths surface. What is the time for this journey?


having trouble with getting the U^2 = gR bit
The question as asked requires us to assume U2 = gR, and then prove from this assumption that the projectile reaches the height R above the Earth's surface.
 
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