# Find a upper bound

• ZuzooVn

#### ZuzooVn

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

Thanks

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?

Hi ZuzooVn! Welcome to PF!

ok … draw y = sin(x).

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

what do you get?

Nope!

Just do it!

Nope!

Just do it!

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

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

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

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.

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

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?

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

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.