Need some serious help with this one problem from my HW:

a) If f(x) = sin(sinx), use a graph to find an upper bound for |f^(4) (x)| (Note that this is not to the 4th power, I believe it means the 4th derivative)

Use Simpsons Rule with n = 10 to approximate the integral of f(x)dx from 0 to pi and use part a) to estimate the error

c) How large should n be to guarantee that the size of the error in using Sn is less than 0.00001


Let me know if you can help me complete a, b, or c.


Thanks for looking guys!

a )
If I do no mistake, f^(4)(x) = cos(sin x).sin x.(4cos² x + 1) + sin(sin x).(cos^4 x + 3cos² x - 1)
which leads to a "brutal" upper bound of 8 for |f^(4)(x)|.
Plotting the graph seems to give you a finest majoration : |f^(4)(x)| < 3
(actually you could as well take 2.626)

b )
What kind of Simson's rule do you want to use ?
with n = 10,
ai = 0 + (pi - 0) * i / 10 = i*pi / 10

Integral from 0 to pi of f ~ (1/30) * ( f(x0) + 4.f(x1) + 2.f(x2) + 4.f(x3) + 2.f(x4) + ... + 4.f(x9) + f(x10) )
~ (1/30) * ( f(0) + 4.f(pi/10) + 2.f(pi/5) + 4.f(3pi/10) + ... + 4.f(9pi/10) + f(pi) )

I leave you the computation.

|error| <= (1/180) * (1/n^4) * (pi - 0) * max | f^(4)|
|error| < (1/1800000) * pi * 3
|error| < 0.0000053

c )
With the above method :

(1/180) * (1/n^4) * pi * 2.626 < 0.00001

solve for n : n = 9.

Seems I made a mistake on the fourth derivative :

f^(4)(x) = cos(sin x).sin x.(6cos² x + 1) + sin(sin x).(cos^4 x + 7cos² x - 3)

An upper bound for |f^(4)(x)| is 3.77.

Good luck.

tyvm feanur! I really appreciate it!!


Can anyone verify his work or tell if it appears to be correct?


-thanks again guys!