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points of inflexion and horizontal p.o.i (1 Viewer)
- Thread starter izzy_xo
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TheStallion
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Why is a HSC 2011'er thinking they know more than an engineering student?
+1
and back to the topic..
at first i got confused as well by these 2 things...
horizontal pt of inflexion is when y' = 0, y" = 0
when the inflexion pt is at the stationary pt, it is a horizontal pt of inflexion..
that how i remember it. :S
but can anyone tell me whats the purpose of the "horizontal" point of inflexion please?
thanks
annabackwards
<3 Prophet 9
I don't think there's a purpose, it's just another term for a feature you need to be aware of.
y' = dx/dy or the 1st derivative.
y'' = dx^2/d^2y or the 2nd derivative,
Yes, it's refering to how you find points of inflection and stationary points, which you do by letting y' = 0 for stat pts and y'' = 0 for POIs (of course you need to prove concavity change too).
They have to do with a Horizontal POI because you need to find horizontal POIs by showing that they both satisfy the requirements of y' = 0 and y'' = 0
so does that mean, when the x value is subbed in y' and y'', y' and y'' both =0?I don't think there's a purpose, it's just another term for a feature you need to be aware of.
y' = dx/dy or the 1st derivative.
y'' = dx^2/d^2y or the 2nd derivative,
Yes, it's refering to how you find points of inflection and stationary points, which you do by letting y' = 0 for stat pts and y'' = 0 for POIs (of course you need to prove concavity change too).
They have to do with a Horizontal POI because you need to find horizontal POIs by showing that they both satisfy the requirements of y' = 0 and y'' = 0
annabackwards
<3 Prophet 9
If it's a Horizontal POI, yes
Recap: A point of inflection of a nice function y is simply a minimum or maximum of y'.
You already know how to find mins and maxs of nice functions, so you just apply the same knowledge to the function y'. That is, you look for points x such that y''(x) = 0 (at this stage x is a potential point of inflection of y), and then you test to see of x is a min or max of y' by (1) applying the second derivative test to y' at the point x, or (2) by substituting x-values either side of your potential point of inflection into y' to get a feel for what the function is doing near your potential point of inflection.
Method 2 is inherently dodgy since it doesn't prove anything on its own, but will probably be good enough for any function you're likely to see and any examiner you're likely to put a test in front of. Except in special cases, method 1 has the potential to actually prove that your potential point is a point of inflection. The special cases for which this method doesn't work, and certain proof one way or the other on whether your potential point of inflection is a point of inflection, are easily taken care of by a simple generalisation of the second derivative test, but this generalisation is not considered 'HSC knowledge' (though it is a simple consequence of it).
In the special case that your point of inflection x satisfies y'(x) = 0 it's called a horizontal point of inflection, simply because the tangent line to the function y at the point x is horizontal. If you would like a more 'geometric' feeling for a horizontal point of inflection, it is not just a point where the function 'slows down' and 'speeds back up' or vice versa (a point of inflection), it's a point where the function actually slows down to a complete stop before continuing to increase or decrease as it was before. If you want pictures to help with that, the functions f(x) = x^3 and g(x) = sin(x) at the point x=0 should help you out.
You already know how to find mins and maxs of nice functions, so you just apply the same knowledge to the function y'. That is, you look for points x such that y''(x) = 0 (at this stage x is a potential point of inflection of y), and then you test to see of x is a min or max of y' by (1) applying the second derivative test to y' at the point x, or (2) by substituting x-values either side of your potential point of inflection into y' to get a feel for what the function is doing near your potential point of inflection.
Method 2 is inherently dodgy since it doesn't prove anything on its own, but will probably be good enough for any function you're likely to see and any examiner you're likely to put a test in front of. Except in special cases, method 1 has the potential to actually prove that your potential point is a point of inflection. The special cases for which this method doesn't work, and certain proof one way or the other on whether your potential point of inflection is a point of inflection, are easily taken care of by a simple generalisation of the second derivative test, but this generalisation is not considered 'HSC knowledge' (though it is a simple consequence of it).
In the special case that your point of inflection x satisfies y'(x) = 0 it's called a horizontal point of inflection, simply because the tangent line to the function y at the point x is horizontal. If you would like a more 'geometric' feeling for a horizontal point of inflection, it is not just a point where the function 'slows down' and 'speeds back up' or vice versa (a point of inflection), it's a point where the function actually slows down to a complete stop before continuing to increase or decrease as it was before. If you want pictures to help with that, the functions f(x) = x^3 and g(x) = sin(x) at the point x=0 should help you out.
I don't. I just knew that this engineer was wrong in this particular instance.
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