Third derivative method (1 Viewer)

[hyparzero]

Its a good method, but its a shame that many schools will claim it cannot be used unless first proven first - which is a bummer if theres only 1 or 2 inflexion questions in the exam.

I totally disagree with this policy which forbids the use of non-syllabus knowledge. For example, a difficult One Dimensional Motion question in 4U could be solved using pages pure calculus or you can simply using a laGrangian mechanics theorem to solve it in 2 lines - but it is unlikely you'll get marks for it if you don't prove it.

 

[buchanan]

I totally disagree with this policy which forbids the use of non-syllabus knowledge.

Yeah. The syllabus is only supposed to show the minimum that is required. It's not supposed to be abused in such a way as to act like a tool with which to exclude all other knowledge. Such abuse leads to mediocrity.

The Stage 6 syllabuses will be changed soon and so I've made a submission for it here. I've made it clear in there what role the syllabus has - and it isn't a sentinel for mediocrity.

 

[buchanan]

Someone asked in another thread about inflections for f(x)=x<SUP>9</SUP>/(1+x).

It's easy. Use Maclaurin series and 9th derivative test.
This would be far more elegant than the syllabus method.

f(x)=Σ(-1)<SUP>n</SUP>x<SUP>n+9</SUP>. So if f''(x)=0 then x=0.

f<SUP>(n)</SUP>(0)=0 for n=0,1,2,...,8, but f<SUP>(9)</SUP>(0)=9!&ne;0, so (0,0) is a horizontal inflection.

Whoever constructed this question thought it would work against me. I turns out it has worked in my favour. Thanks for that!

 

[hyparzero]

Someone asked in another thread about inflections for f(x)=x⁹/(1+x).

It's easy. Use Maclaurin series and 9th derivative test.
This would be far more elegant than the syllabus method.

f(x)=Σ(-1)ⁿxⁿ⁺⁹. So if f''(x)=0 then x=0.

f⁽ⁿ⁾(0)=0 for n=0,1,2,...,8, but f⁽⁹⁾(0)=9!*0, so (0,0) is a horizontal inflection.

Whoever constructed this question thought it would work against me. I turns out it has worked in my favour. Thanks for that!

Out of 7 maths teachers, only 1 knew what a Maclaurin Series is

Also, buchanan, can you use the maclaurin series to find inflexion points for more difficult functions like the one you shown above if the third derivative method fails for some questions?

 

[acmilan]

Out of 7 maths teachers, only 1 knew what a Maclaurin Series is

erm..are you serious? How do they become math teachers?

 

[buchanan]

Sometimes Maclaurin series expansion makes a problem easier. All I did was get the expansion Σ(-1)<SUP>n</SUP>x<SUP>n</SUP> for 1/(x+1) and then multiply by x<SUP>9</SUP> and went from there. It wasn't hard and you don't have to be a genius to use it. Here's more info:

http://mathworld.wolfram.com/MaclaurinSeries.html

 

[hyparzero]

erm..are you serious? How do they become math teachers?

I have no freaking idea.. heres an example of what one my teachers did...

1. Find the intersection of the following 2 curves:
y²=x³

and y²=2 - x


My teacher basically had no idea and drew up a table of values for both curves and went: "aha! these two curves have the same value!"

iwas like omfg

 

[buchanan]

Find the intersection of the following 2 curves:
y²=x³

and y²=2 - x

Do you mean your teacher can't solve cubics either?

x<SUP>3</SUP>+x-2=0 so test divisors of 2, in particular, 1 works, so
x<SUP>3</SUP>+x-2=(x-1)(x<SUP>2</SUP>+x+2)=0 and the discriminant for the second factor is -7 so x=1 and the points are (1,1), (1,-1). If no divisor worked you could use the cubic formula.

 

[hyparzero]

Yup, my teacher's "table of values" method only found (1,1), but missed (1,-1)

totally ridiculous

 

[hyparzero]

buchanan, whats the general maclaurin series expansion for functions?

my exercise book says:

f(x) = Σ f^(k)(c)(x - c )^k / k!

where c is the center of convergence. Care to help on how to apply this properly?

MathWorld shows some explicit maclaruin forms of common functions, do you just remember these off by heart?

 

[buchanan]

f(x) = Σ f^(k)(c)(x - c )^k / k!

where c is the center of convergence.

That's Taylor's series which is a bit more complicated:
http://mathworld.wolfram.com/TaylorSeries.html

Maclaurin has c=0. There's no need to remember them off by heart.

As for your teacher's problem, a graphical approach would clearly indicate there should be 2 solutions, so I'd say your teacher was either a bit careless at the time (which is forgiveable) or perhaps didn't know quite what they were doing (which isn't):

Green curve: y²=2-x

Red curve: y²=x³

 

[buchanan]

"Ideally, when teaching any course, we would like our students to be able to take the ideas they have learned and adapt them to other situations. Having a student develop a third derivative test for possible points of inflection and apply it intelligently certainly fits that goal."

Gordon, S. P., "New" and "Old" Calculus: Student Reactions and Comments", Focus on Calculus, Calculus Consortium, Harvard University, Winter 1997, Issue No. 12.
------------------------------

With a comment like that published by the world's best university in support of my approach, we don't need dissenting voices coming from UNSW philistines attempting to curb my enthusiasm (......again......). There is just one thing we can conclude about such attempts. IT ISN'T WORKING!

 

[kami]

The minimum requirements for content knowledge set by NSW DET for registration as a maths teacher are very low - you only need to have studied maths to 2nd year level. Many of the uni students who post on these forums (you included, I think) would have already surpassed these requirements.

Actually its even lower than that, only first year maths in either pure or applied is required to teach maths as your second teaching area at least thats what TEACH.NSW's site says. Which is truly frightening when you say that many teachers are not truly qualified to teach - does this mean they have not even studied to completion at first year level?

 

[Raginsheep]

Of course the reverse is true as well. Being well knowledged but having no ability to convey that knowledge is not good either.

 

[Raginsheep]

I've had certain teachers who "was part of the committee which wrote the syllabus" who were complete failures at teachers while I've learned quite well from teachers who's knowledge was far more limited.

Saying that though, having a teacher who is only "one chapter ahead" is completely unacceptable.

 

[Rax]

Yeah it is a bit of a problem, luckily for me my teachers have been pretty competent.

However some of my friends currently in year 9 and year 10 struggle with maths and when I explain some things to them they say stuff like "wow thats so logical" which makes me a bit worried why the teachers coudnt do it.......and I am talking about simple algebra here, nothing too crazy