Question

Simpson's Rule error checking? Help with finding K?

Answer 1

I need help with part b, I have the equation for error

Answer 2

The approximation I got for the first part is 19.20666667

Answer 3

Do you multiply?

Answer 4

Multiply what?

Answer 5

Mmm I'm not really familiar with Simpson's Rule..
I found this on Wikipedia for calculating error of Simpson's Rule,\[\Large\bf\sf \frac{1}{90}\left(\frac{b-a}{2}\right)^{5}\left|f^{(4)}(\xi)\right|,\qquad\qquad \xi\in\left[a,b\right]\]Does that formula look familiar maybe? :\

Answer 6

Yeah, that's a non-simplified version of\[\frac{ K(b-a)^3 }{ 180n^4 }\]

Answer 7

K and n? :x

Answer 8

I know for this that n=8 and a=0 and b=1.6

Answer 9

K should be the absolute max for the 4th derivative of g(x)

Answer 10

I think n refers to how many parts the interval is meant to be split up

Answer 11

For our K, where do we evaluate g(x) at?
Do we choose an arbitrary value in the region?
Or the maximum value in the region? Or zero? D: Hmmm

Answer 12

I have no clue, I guess g(x) is some random function that's left unknown because of the table

Answer 13

\[\Large\bf\sf \frac{1.6^3}{180\cdot8^4}\cdot K\quad=\quad \frac{1.6^3}{180\cdot8^4}\cdot \text{max}_{g^{(4)}\in[-5,2]}\left|g^{(4)}(x)\right|\]

And then we just see which gives us the largest value I guess?
Which is clearly going to be the -5 right?

Answer 14

\[\Large\bf\sf =\frac{1.6^3}{180\cdot8^4}\left|-5\right|\]

Answer 15

hmm

Answer 16

So the absolute value of -5?

Answer 17

My friend hinted that K=5 but didn't show me how to prove it

Answer 18

Taking the absolute value of -5 gives us 5, so that gives us the largest value for our error... whichhhh I'm assuming is probably what we want. Hmm

Answer 19

Do you have an answer key to check against? :D

Answer 20

How did you get -5?

Answer 21

For \(\Large\bf\sf 0\le x\le1.6\) they told us that our 4th derivative of g is restricted to

\(\Large\bf\sf -5\le g^(4)(x)\le 2\)

Answer 22

Woops I messed up my exponent :P\[\Large\bf\sf g^{(4)}(x)\]

Answer 23

So our g's are stuck between -5 and 2, we just want the largest absolute in that range ( I think ).

Answer 24

The answer given is the error should be less than 0.000071111

Answer 25

And the powers for the formula I gave are supposed to be 5 on top and 4 on bottom, I made a mistake there

Answer 26

Oh like this? :o\[\Large\bf\sf \frac{1.6^5}{180\cdot8^4}\cdot K\]

Answer 27

Yeah

Answer 28

OOo good the 5 is working ! \c:/

Answer 29

I think that's just what I'm looking for :)

Answer 30

\[\Large\bf\sf \frac{1.6^5}{180\cdot8^4}\cdot |-5|\]

Answer 31

Multiply the numbers?