Advanced Higher Maths 2011-2012 : Discussion and Help

Original post by JaggySnake95

Is it something to do with deacceleration and acceleration near the stationary points? :s-smilie:

Sort of! Here's a plot of a function, its derivative and its second derivative f''(x). Stationary points are where f'(x) - rate of change - is zero. The function is locally unchanging.

But what about the nature of those stationary points? Well, if f''(x) is positive, then f'(x) is climbing as you increase x, which means that f(x) is climbing faster than it is right now as you increase x. Which means (look at the plot above) that if f'(x) = 0 and f''(x) > 0 then we've got a minimum.

Similarly, if f'(x) = 0 and f''(x) < 0 then we've a maximum. f''(x) = 0 is slightly messier - we'll have to check the sign just to the left and right of x to see.

Reply 981

JordanR

12

I actually used the second derivative test in Higher -- it worked well for me. Was much faster than drawing those pesky nature tables.

Reply 982

I'm not entirely sure why it's not more generally taught, to be honest. It's fairly intuitive if you draw a couple graphs and spend longer than the 2 minutes I did thinking about how to describe it. And, like you say, faster and more elegant.

Reply 983

mimx

Original post by JordanR

I actually used the second derivative test in Higher -- it worked well for me. Was much faster than drawing those pesky nature tables.

Yeah I have no idea why they don't teach it since it's literally half the work/time and you might have more than one question per paper where it's of use. I was actually told not to use it. :rolleyes:

Did it anyway of course, it was the height of rebellion. :cool:

Original post by TheUnbeliever

I'm not entirely sure why it's not more generally taught, to be honest. It's fairly intuitive if you draw a couple graphs and spend longer than the 2 minutes I did thinking about how to describe it. And, like you say, faster and more elegant.

To be fair my Higher class had to have 5 revisions on multiplying and dividing fractions, it was pretty dire. TSR is way above average in maths ability.

(edited 12 years ago)

Reply 984

Original post by mimx

To be fair my Higher class had to have 5 revisions on multiplying and dividing fractions, it was pretty dire. TSR is way above average in maths ability.

Trust me, I know :p:

Heck, my class had 24 A1s and 2 A2s, and the year overall had 66% A grades... That said, I still don't see that the second derivative test is any more complicated. To get to that point, you must have already differentiated once so can hopefully repeat the process - and the checking of the sign is, at worst, exactly as complex as the nature table. :confused:

Reply 985

Original post by JordanR

I actually used the second derivative test in Higher -- it worked well for me. Was much faster than drawing those pesky nature tables.

Original post by TheUnbeliever

I'm not entirely sure why it's not more generally taught, to be honest. It's fairly intuitive if you draw a couple graphs and spend longer than the 2 minutes I did thinking about how to describe it. And, like you say, faster and more elegant.

Original post by mimx

Yeah I have no idea why they don't teach it since it's literally half the work/time and you might have more than one question per paper where it's of use. I was actually told not to use it. :rolleyes:

Did it anyway of course, it was the height of rebellion. :cool:

It's because it could fail (when it evaluates to zero) - and very few people, even at AH, knows 1) it is a test failure which tells you nothing (and possibly not a point of inflection!) and 2) how to handle it correctly when it does fail. Just try using it on y = x^4 for example.

(edited 12 years ago)

Reply 986

Slumpy

I quite like the nature table. I'm not sure why though, analysis should've put me off it for life!

Reply 987

JordanR

12

I have no idea either. It's just counter-intuitive not to teach it. At least teach both and provide the option of either...

My Higher class wasn't wonderful either, though. Not that I was much better.

Reply 988

Original post by ukdragon37

It's because it could fail (when it evaluates to zero) - and very few people, even at AH, knows 1) it is a test failure and not a point of inflection and 2) how to handle it correctly when it does fail. Just try using it on y = x^4 for example.

Sure, but in that worst case you can resort to the nature table - which leaves you no worse off than you were already, but better in the typical case.

Reply 989

Original post by JordanR

At least teach both and provide the option of either...

Recipe for trashing your (as a teacher/school) results as you confuse the marginal students.

Reply 990

Original post by TheUnbeliever

Sure, but in that worst case you can resort to the nature table - which leaves you no worse off than you were already, but better in the typical case.

Original post by JordanR

I have no idea either. It's just counter-intuitive not to teach it. At least teach both and provide the option of either...

Original post by TheUnbeliever

Recipe for trashing your (as a teacher/school) results as you confuse the marginal students.

For that exact reason I think they would rather teach a single longer but fool-proof method. You can use it if you are clever and are aware of the consequences, but teaching it is another matter.

(edited 12 years ago)

Reply 991

Original post by Slumpy

I quite like the nature table. I'm not sure why though, analysis should've put me off it for life!

I would like to see someone drawing a nature table for the Weierstrass function :smug:

Reply 992

Original post by ukdragon37

For that exact reason I think they would rather teach a single longer but fool-proof method. You can use it if you are clever and are aware of the consequences, but teaching it is another matter.

But it's entirely possible to codify the second derivative test exactly as a 'longer (in the worst case) but fool-proof method': you just subsume the nature table as the process if f''(x) = 0. My complaint is saying that "you can do X, or Y - which might sometimes involve X as well" is confusing in a way that "you can do X, with means doing O in case A, Q in case B and R in case C" isn't. (IMHO, of course.)

Reply 993

Slumpy

Original post by ukdragon37

I would like to see someone drawing a nature table for the Weierstrass function :smug:

I prefer the analysis answer to everything: xcos(1/x)...or should that be cos(1/x)/x? Pleh, one of those!

Reply 994

Original post by Slumpy

cos(1/x)/x

That is one ugly-ass* function.

* technical term.

Reply 995

mimx

Original post by TheUnbeliever

Sure, but in that worst case you can resort to the nature table - which leaves you no worse off than you were already, but better in the typical case.

Typically it won't be a full nature table anyway, usually with what they give you in Higher just one of your SP checks will fail and you'll check either side for that one point and whizz through the others that do work out.

I just really hate the nature table I guess, it makes an otherwise elegant solution kind of ham fisted. Like sketching a graph by plugging in x=1, x=2, x=3 etc.

(edited 12 years ago)

Reply 996

Slumpy

Original post by TheUnbeliever

That is one ugly-ass* function.

* technical term.

I think you mean 'babe of a function'. And I think xcos(1/x) is the answer to everything anyways :p:

Reply 997