how many times can we differentiate this function??

[transgalactic]

how many times can we differentiate this splitted function on point x=0

??

[math]

f(x) = \{ x^{2n} \sin (\frac{1}{x})

,x \ne 0

[/math]

 

i know i need to prove it by induction

[math]\dfrac{\mathrm{d}^r}{\mathrm{d}x^r}\big(x^{2n}\sin(x^{-1})\big)=

\begin{cases}

(-1)^rx^{2(n-r)}\sin(x^{-1}+\frac{1}{2}r\pi)+x^{2(n-r)}g_{n,r}(x) & x\neq 0\\

0& x=0

\end{cases}

[/math]

for all r≤n, where g_{n,r}(x) is continuous at 0 and vanishes there.

what to do next??

[Bignose]

So,

 

1) I moved a bunch more of these threads from Math to Homework, since you seem to be constantly posting on how to do these problems, may I suggest you continue to post in Homework when you need help.

 

2) I am hoping that this doesn't come off too snide or rude or anything, because I really don't mean it to sound that way: But, is there not a TA or professor's office hours you can visit to get the help you seem to need with these? Because you've asked a lot of questions that are pretty similar, and if you continuously struggle with these, I think that talking with your instructor would be much, much more fruitful than asking a forum.

[Severian]

Haven't you answered your own question already? Your can be differentiated (at x=0) n times. Once you get to r=n, it is no longer continuous since

[math]\lim_{x \to 0} \dfrac{\mathrm{d}^n}{\mathrm{d}x^n}\big(x^{2n}\sin(x^{-1})\big)=(-1)^n \lim_{x \to 0} \sin \left(x^{-1}+\frac{1}{2}n\pi \right) \neq 0[/math]

[transgalactic]

how many times can we differentiate this splitted function on point x=0

??

[math]

f(x) = \{ x^{2n} \sin (\frac{1}{x})

,x \ne 0

[/math]

 

i know i need to prove it by induction

[math]\dfrac{\mathrm{d}^r}{\mathrm{d}x^r}\big(x^{2n}\sin(x^{-1})\big)=

\begin{cases}

(-1)^rx^{2(n-r)}\sin(x^{-1}+\frac{1}{2}r\pi)+x^{2(n-r)}g_{n,r}(x) & x\neq 0\\

0& x=0

\end{cases}

[/math]

for all r≤n, where g_{n,r}(x) is continuous at 0 and vanishes there.

what to do next??

 

this is not an induction

what is the full proof?

[D H]

I'm going to be blunt, transgalactic: Do it yourself. You are using as a crutch homework help sites such as this, along with the plethora of other sites where you post the exact same questions. As a consequence, while you might be getting your homework done, you obviously are not learning a blasted thing.

 

How are you going to fare when it comes to exam time? How are you going to fare in upper level courses where there aren't so many people around who can help? How are you going to fare in the real world where the people who can help work for companies that compete with the one for which you work?

[Bignose]

I'm going to be blunt, transgalactic: Do it yourself. You are using as a crutch homework help sites such as this, along with the plethora of other sites where you post the exact same questions. As a consequence, while you might be getting your homework done, you obviously are not learning a blasted thing.

 

How are you going to fare when it comes to exam time? How are you going to fare in upper level courses where there aren't so many people around who can help? How are you going to fare in the real world where the people who can help work for companies that compete with the one for which you work?

 

I concur completely. This is the main reason I've haven't participated much in these threads. Because it is obvious to me as well that you aren't really learning anything either. So, I'm with D H: do it yourself, or at the very least go and have a long sit-down with your TA and/or professor.