Find Upper Bound for abs(f(4)(x)) of f(x)=sin(sin(x))

[ZuzooVn]

help me, please

if f(X) = sin(sin(x)), use a graph to find a upper bound for abs(f⁽⁴⁾(x))


Thanks

 

[tiny-tim]

Welcome to PF!

Hi ZuzooVn! Welcome to PF!

if f(X) = sin(sin(x)), use a graph to find a upper bound for abs(f⁽⁴⁾(x))

ok … draw y = sin(x).

Now turn the paper sideways and draw x = sin(y) …

what do you get?

 

[ZuzooVn]

Hi ZuzooVn! Welcome to PF!


ok … draw y = sin(x).

Now turn the paper sideways and draw x = sin(y) …

what do you get?

Would u please tell me more detail about your solution?

 

[tiny-tim]

Would u please tell me more detail about your solution?

Nope!

Just do it! ​

 

[ZuzooVn]

Nope!

Just do it! ​

Please

Because, i didn't know how to find the upper bound

 

[HallsofIvy]

tiny-tim has suggested a first step. Have you done it yet?

 

[ZuzooVn]

tiny-tim has suggested a first step. Have you done it yet?

yes, i have done it .

But because I'm a Vietnamese, so my English skill isn't good :D

 

[HallsofIvy]

Excellent! Thank you.

Now, tiny-tim, what in the world are you talking about? I'm afraid I dont' see your point either.

I would probably use "brute strength"

If y= sin(sin(x)), then y'= -cos(sin(x))(-cos(x))= cos(x)cos(cos(x)). Now, instead of actually doing the other derivatives (because they get really messy!), use the fact that the nth derivative of (f(x)g(x)) will be $\sum _nC_i f^{i}g^{n-i}$ to see that we will, after three more derivatives, have a sum of 4 terms with binomial coeficients times sin and cos- and the largest possible value for sine or cosine is 1.

 

[AUMathTutor]

Unless I made a silly mistake typing things in, it appears that Wolfram Alpha thinks it should be around 3.76.

 

[mXSCNT]

help me, please

if f(X) = sin(sin(x)), use a graph to find a upper bound for abs(f⁽⁴⁾(x))


Thanks

You need to define what f⁽⁴⁾(x) means. Do you mean, the fourth iteration of f on x, i.e. f o f o f o f (x)? Or do you mean (as others have interpreted) the fourth derivative of f?

 

[ZuzooVn]

You need to define what f⁽⁴⁾(x) means. Do you mean, the fourth iteration of f on x, i.e. f o f o f o f (x)? Or do you mean (as others have interpreted) the fourth derivative of f?

I means the fourth derivative of f

 

[HallsofIvy]

Unless I made a silly mistake typing things in, it appears that Wolfram Alpha thinks it should be around 3.76.

Do you mean least upper bound? I get 8 as an upper bound.