How to get better at maths.

Scroll to see replies

Reply 80

DFranklin

Original post by StarvingAutist

Oh, um, wow :mmm:

that was sure convincing..
Maybe I'm just deluded then :lol:

But why don't they include it? The space taken up by that table & paragraph could easily have been used :eek:

How would you, say, prove that

\dfrac{d}{dx} x^{0.3} = 0.3 x^{-0.7}

Reply 81

StarvingAutist

Original post by DFranklin

How would you, say, prove that

\dfrac{d}{dx} x^{0.3} = 0.3 x^{-0.7}

Prove the case for xⁿ and shove in 0.3 :wink:

edit: in case you wanted a better answer (though with shoddy notation)

(edited 9 years ago)

Webcam-1417637653.jpeg26.1KB

Reply 82

ThatPerson

Original post by StarvingAutist

Oh, um, wow :mmm:

that was sure convincing..
Maybe I'm just deluded then :lol:

But why don't they include it? The space taken up by that table & paragraph could easily have been used :eek:

They hadn't introduced the binomial theorem at that point, so they couldn't give a proof of it.

Reply 83

DFranklin

Original post by StarvingAutist

Prove the case for xⁿ and shove in 0.3 :wink:

edit: in case you wanted a better answer (though with shoddy notation)

You've used the binomial theorem (in the case of a non-integral power). The standard proof of this requires knowledge of the derivative of x^0.3, so your argument is circular.

Reply 84

StarvingAutist

Original post by DFranklin

You've used the binomial theorem (in the case of a non-integral power). The standard proof of this requires knowledge of the derivative of x^0.3, so your argument is circular.

Damn it, I forgot that. How retarded. In that case, I don't know.

Reply 85

DFranklin

Original post by StarvingAutist

Damn it, I forgot that. How retarded. In that case, I don't know.

It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).

Reply 86

davros

Original post by StarvingAutist

Damn it, I forgot that. How retarded. In that case, I don't know.

Original post by DFranklin

It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).

More fundamentally, it's no good talking about the derivative of a function until you can actually say how the function itself works.

How do you know what

x^{0.3}

means for a real number x? OK, this isn't too bad because 0.3 is rational, so we can argue (correctly) that it's the 10th root of x cubed. But what do we do when confronted by

x^{\sqrt{2}}

x^r

for general real r?

There are good reasons why Bostock and Chandler (and comparable texts) say "take this [result] on trust" when introducing rules for things like differentiation :smile:

Reply 87

StarvingAutist

Original post by DFranklin

It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).

How do you do it? Use xⁿ = e^ln(^x^{^n)}= e^nlnx and prove the chain rule? How much more is there?

Reply 88

StarvingAutist

Original post by davros

More fundamentally, it's no good talking about the derivative of a function until you can actually say how the function itself works.

How do you know what

x^{0.3}

means for a real number x? OK, this isn't too bad because 0.3 is rational, so we can argue (correctly) that it's the 10th root of x cubed. But what do we do when confronted by

x^{\sqrt{2}}

x^r

for general real r?

There are good reasons why Bostock and Chandler (and comparable texts) say "take this [result] on trust" when introducing rules for things like differentiation :smile:

Yeah, very true... I suppose it's a good thing I'm going for physics; it seems I'd fail a maths degree! It's all interesting though, and things just get deeper the harder you look :redface:

Reply 89

DFranklin

Original post by StarvingAutist

How do you do it? Use xⁿ = e^ln(^x^{^n)}= e^nlnx and prove the chain rule? How much more is there?

Also need to define e^x and ln x, show that the definitions do what you expect and show that defining x^n as you describe does what you expect.

Edit: The last bit includes what davros was alluding to when talking about x^pi etc.

(edited 9 years ago)

Reply 90

davros

Original post by StarvingAutist

How do you do it? Use xⁿ = e^ln(^x^{^n)}= e^nlnx and prove the chain rule? How much more is there?

What are you asking about now?

If you're wanting to differentiate

x^r

you can define the exponential function (and prove that it has an inverse, ln), and then define

x^r = exp(rlnx)

, using the differentiability of the exponential function (which you also need to prove!).

If you're talking about proving the binomial expansion for general exponent, again you have to define what a power series is, define radius of convergence, prove that differentiability makes sense within the radius of convergence and THEN use the limit definition of the derivative to show that the power series expansion gives you the function you started with!

Reply 91

StarvingAutist

Original post by DFranklin

Original post by davros

What are you asking about now?

If you're wanting to differentiate

x^r

you can define the exponential function (and prove that it has an inverse, ln), and then define

x^r = exp(rlnx)

Oh jesus, what a can of worms I opened! I can see why no-one bothers to prove it now :lol:

I suppose doing what I did is just as unsatisfactory as saying 'trust me, this works' :colondollar:

Reply 92

Ngombulango

Once you've decided,absolutely you will. Mathematics is an anyday carear with lot of trials. Just be focused.

Reply 93

tombayes

Original post by AkashdeepDeb

Follow these 3 steps, they really do work!

1) Never miss a single lecture, always attend one even if the topic taught is easy.
2) Before attending a class, always get to know what is being taught, ie. learn a sizable chunk of material prior to the lecture, so that you know what you are going to learn; this boosts up confidence.
3) Get more books and just practice. Practice, practice and practice.

Good luck!

Ideally yes but certainly (2) requires extreme dedication - I never did this and was fine - don't overwork - this is extremely important in maths.

for 1) in first year I attended around 75% of lectures but its been increasing every year!

3) practice very true