Question

Calculus.

Answer 1

@mathmale I used trapezoid and then did it regularly and got the same answer so it has to be b or d. I don't know how to use simpson's rule.

Answer 2

@pooja195 @jim_thompson5910 @zepdrix @jhonyy9

Answer 3

Hopefully this page will help

http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

Answer 4

focus on the part boxed in red (see attached)

Answer 5

If I did it right I got b.

Answer 6

well first tell me what f ' (x) is

f(x) = 3x+3
f ' (x) = ???

Answer 7

the derivative is 3 due to the power rule.

Answer 8

so if f ' (x) = 3, then what is f '' (x) equal to?

Answer 9

0

Answer 10

So if
\[\Large \Big|f ''(x)\Big| \le K\]
then
\[\Large \Big|0\Big| \le K\]
the smallest K can be is 0

Answer 11

right so that is in our b answer and our d. So we find the max it could be?

Answer 12

What does this mean? K = 0 means that there is no error for the trapezoid rule.

Think about the graph of f(x) = 3x+3. This is a linear equation. The area from x = a to x = b can be EXACTLY found using a single trapezoid. There is no error.

Answer 13

Right I already got that part. The part I am confused on is the Simpson's Rule part.

Answer 14

Right. I had narrowed it down to b and d, but I don't know the second part of the question the simpsons rule part.

Answer 15

what is \(\Large f^{(4)}(x)\) equal to?

this is the 4th derivative

NOT the 4th power

Answer 16

of our original f(x)=3x+3?

Answer 17

yeah

Answer 18

Still 0

Answer 19

1st=3 2nd=0 3rd=0 4th=0.

Answer 20

correct, hold on let me see if I can find a simpsons rule calculator real quick

Answer 21

ok. Thanks for the help really appreciate it :)

Answer 22

ok I had to confirm the answer, but it's definitely 0 as well

this page

http://mathworld.wolfram.com/SimpsonsRule.html

says `Simpson's rule actually gives EXACT RESULTS when approximating integrals of polynomials up to cubic degree.`

I put EXACT RESULTS in all caps to emphasize (the original article has it all in lowercase)

so that's why the error is 0. f(x) in this case is a linear function which has degree less than 3

Answer 23

ok thanks. I have one last question I am going to tag you and close this one.

Answer 24

The way I read your original problem, you need to find the exact area under the curve between the given limits, and then to approx. that area using n=4 using the Trap. Rule and then Simpson's Rule.

That way, you'll have two figures to compare to the exact area.

You do need to know the precise formula for both the Tripez. Rule and Simpson's Rule, so why not look them up yourself and write them down for future reference?