Picture Problem using Simpson's Rule
Using Simpsons Rule (n = 8) , approximate an integral

- blackstreet23

Picture Problem using Simpson's Rule
Using Simpsons Rule (n = 8) , approximate an integral

- chestercat

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- blackstreet23

Where can I post pictures ?

- blackstreet23

##### 1 Attachment

- nincompoop

http://www.intmath.com/integration/6-simpsons-rule.php

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## More answers

- SolomonZelman

\(\large\color{black}{ \displaystyle {\rm Simpson's~Rule} \\[0.9em] {\rm Area~~of~f~~over~~[1,3]~~with~n=8}~\\[1.9em]~\displaystyle \normalsize \frac{3-1}{8}\left(f(1)+4f(1.25)+2f(1.50)+4f(1.75)+2f(2) \\ +4f(2.25)+2f(2.50)+4f(2.75)+f(3)\right) }\)

- SolomonZelman

then the maximum error formula is
\(\large\color{black}{ \displaystyle {\rm E}_{S}=\frac{ k(b-a)^5 }{180n^4} }\)
where k is the bound on the 4th derivative.... (if you want this one)

- mathmate

@blackstreet23 @solomanzelman
Do you have everything you need to finish all four parts of the problem?

- blackstreet23

No I am stuck at part b

- mathmate

Did you try @SolomonZelman 's formula? (you must have seen that before)

- mathmate

- blackstreet23

I mean I know that maximum points in a closed interval picture occur at critical points and end points. so do I need to take the fifth derivative ?

- freckles

isn't the 4th derivative given? (you don't need 5th derivative )

- blackstreet23

i need the highest point of the fourth derivative and extrema occur at end points and critical points.

- freckles

I see... The fifth derivative doesn't look too easy to guess highest
you could differentiate the 4th derivative to find the 5th derivative

- blackstreet23

but that is correct right?

- freckles

I see... The forth derivative doesn't look too easy to guess highest
*

- freckles

http://www.wolframalpha.com/input/?i=f%28x%29%3D-3%285x%5E4-60x%5E2%2B10%29%2F%2816%28x%5E2%2B1%29%5E%2815%2F4%29%29+x%3D-10+to+10+and+y%3D-1+to+1 beautiful finally figured out how to play with zooming options

- freckles

notice the max is just under 1
for approximately what x value does that occur
the 4th derivative is even (doesn't matter if you choose the positive or negative version of this particular number )

- blackstreet23

but the fifth derivative is still necessary just to show work right?

- mathmate

- mathmate

I would graph and estimate the max. instead of going through the 5th derivative AND solving the resulting equation for f5(x)=0.
Generally the actual error is very much lower than the upper bound estimate, so numerically, it should be ok.

- mathmate

gtg

- mathmate

Graphically, it looks like to be at x=0.8.
Poking around with a few iterations give x=0.787, and f4(.787) around .7149.
That should solve part D.
For part C, you don't need the relative max.

- blackstreet23

but because those values are not within the interval i do not need to worry of them until part d right?

- blackstreet23

- mathmate

Exactly. For part B, the maximum is at x=1, and it's a strictly decreasing function, so no need to find the relative maximum.

- mathmate

For part D, even a few casual guesses will give you the maximum within 3 decimal digits.
Have you done part C (find the value of n)?

- blackstreet23

yes n=6

- blackstreet23

on d the new value of K4 will be 0.7149345503 right?

- mathmate

For part C, I have 10.1 or 10.2, which is kind of awkward, because it will bring the n=12 when we know almost sure that even n=10 may be good.
Yes, the value of \(k_4\) is 0.7149 at about x=0.79.

- mathmate

Correction:
Yes, I have n=5.713 => 6 for part C.
I must have used the error bound ten times less when I calculated yesterday.

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