Author Topic: Trapezoidal RUle (Read 8322 times) Tweet Share

itswags98 · « **on:** September 21, 2016, 07:41:44 pm »

Heya! Im trying to understand why its 1/4 outside the brackets. Overall im pretty confused about the question as im ussually used to the one application rule

Woops sorry, didnt mean to post as a seperate thread. Comp acting funky

RuiAce · « **Reply #1 on:** September 21, 2016, 08:00:48 pm »

Quote from: itswags98 on September 21, 2016, 07:41:44 pm

Heya! Im trying to understand why its 1/4 outside the brackets. Overall im pretty confused about the question as im ussually used to the one application rule

Woops sorry, didnt mean to post as a seperate thread. Comp acting funky

$\text{They used the generalised trapezoidal rule. Not the standard version.}\\ \int_1^3f(x)dx\approx \frac{h}{2}\left[f(1)+2f(1.5)+2f(2)+2f(2.5)+f(3)\right]\\ \text{Where }h=\frac{b-a}{n}=\frac{3-1}{4}=\frac{1}{2}\\ \text{Where }n\text{ is the number of subintervals}$
$\begin{align*}\int_1^3xe^xdx&\approx\frac{\frac{1}{2}}{2}\left(1e^1+3e^{1.5}+4e^{2}+5e^{2.5}+3e^6\right)\end{align*}$
________________________

$\text{If you used the standard trapezoidal rule, there will be some algebraic manipulation needed.}\\ \text{We must approximate 4 different integrals separately:}\\ \begin{align*}\int_1^{1.5}xe^xdx&\approx\frac{1.5-1}{2}(e^1+1.5e^{1.5})\\\int_{1.5}^2xe^xdx&\approx\frac{2-1.5}{2}(1.5e^{1.5}+2e^2)\\ \int_2^{2.5}xe^xdx&\approx \frac{2.5-2}{2}(2e^2+2.5e^{2.5})\\\int_{2.5}^3xe^xdx&\approx \frac{3-2.5}{2}(2.5e^{2.5}+3e^3)\end{align*}$
$\text{And then we sum it all up to get the result.}$
My prediction is that you got the fraction confused because you forgot about the impact of .5 on the computation.

itswags98 · « **Reply #2 on:** September 21, 2016, 09:43:06 pm »

Quote from: RuiAce on September 21, 2016, 08:00:48 pm

$\text{They used the generalised trapezoidal rule. Not the standard version.}\\ \int_1^3f(x)dx\approx \frac{h}{2}\left[f(1)+2f(1.5)+2f(2)+2f(2.5)+f(3)\right]\\ \text{Where }h=\frac{b-a}{n}=\frac{3-1}{4}=\frac{1}{2}\\ \text{Where }n\text{ is the number of subintervals}$
$\begin{align*}\int_1^3xe^xdx&\approx\frac{\frac{1}{2}}{2}\left(1e^1+3e^{1.5}+4e^{2}+5e^{2.5}+3e^6\right)\end{align*}$
________________________

$\text{If you used the standard trapezoidal rule, there will be some algebraic manipulation needed.}\\ \text{We must approximate 4 different integrals separately:}\\ \begin{align*}\int_1^{1.5}xe^xdx&\approx\frac{1.5-1}{2}(e^1+1.5e^{1.5})\\\int_{1.5}^2xe^xdx&\approx\frac{2-1.5}{2}(1.5e^{1.5}+2e^2)\\ \int_2^{2.5}xe^xdx&\approx \frac{2.5-2}{2}(2e^2+2.5e^{2.5})\\\int_{2.5}^3xe^xdx&\approx \frac{3-2.5}{2}(2.5e^{2.5}+3e^3)\end{align*}$
$\text{And then we sum it all up to get the result.}$
My prediction is that you got the fraction confused because you forgot about the impact of .5 on the computation.

Awesome Explanation. I appreciate it Rui!

BPunjabi · « **Reply #3 on:** September 30, 2016, 03:22:58 pm »

Quote from: itswags98 on September 21, 2016, 07:41:44 pm

Heya! Im trying to understand why its 1/4 outside the brackets. Overall im pretty confused about the question as im ussually used to the one application rule

Woops sorry, didnt mean to post as a seperate thread. Comp acting funky

BTW with the h=b-a/n dont forget to change it to a decimal so when it goes over the fraction it does not become to tricky

RuiAce · « **Reply #4 on:** October 05, 2016, 08:23:23 am »

Quote from: BPunjabi on September 30, 2016, 03:22:58 pm

BTW with the h=b-a/n dont forget to change it to a decimal so when it goes over the fraction it does not become to tricky

What's the problem of fractions in fractions? I see that all the time from 2U onwards. Just draw the line longer if the fraction is to be evaluated later than others.

I find that the decimal point causes more confusion because too many people use "dot" to mean multiplication.

BPunjabi · « **Reply #5 on:** October 06, 2016, 11:18:20 pm »

Quote from: RuiAce on October 05, 2016, 08:23:23 am

What's the problem of fractions in fractions? I see that all the time from 2U onwards. Just draw the line longer if the fraction is to be evaluated later than others.

I find that the decimal point causes more confusion because too many people use "dot" to mean multiplication.

With some calculators it can get messy if they dont put in the parenthesises.

RuiAce · « **Reply #6 on:** October 06, 2016, 11:31:27 pm »

Quote from: BPunjabi on October 06, 2016, 11:18:20 pm

With some calculators it can get messy if they dont put in the parenthesises.

That might be a safety precaution, though I never found that a problem. The calculator makes it clear which lines are longer than others. If it helps you, go for it.

BPunjabi · « **Reply #7 on:** October 06, 2016, 11:32:26 pm »

Quote from: RuiAce on October 06, 2016, 11:31:27 pm

That might be a safety precaution, though I never found that a problem. The calculator makes it clear which lines are longer than others. If it helps you, go for it.

True True, what Calculator did you use?

RuiAce · « **Reply #8 on:** October 06, 2016, 11:35:37 pm »

Quote from: BPunjabi on October 06, 2016, 11:32:26 pm

True True, what Calculator did you use?

Casio fx-100 AU PLUS

The display is similar to Casio fx-82 AU PLUS, which is what's often sold to students these days I think

BPunjabi · « **Reply #9 on:** October 06, 2016, 11:37:24 pm »

Quote from: RuiAce on October 06, 2016, 11:35:37 pm

Casio fx-100 AU PLUS

The display is similar to Casio fx-82 AU PLUS, which is what's often sold to students these days I think

When you walk in do they check what calculator your using?

RuiAce · « **Reply #10 on:** October 06, 2016, 11:39:33 pm »

Quote from: BPunjabi on October 06, 2016, 11:37:24 pm

When you walk in do they check what calculator your using?

Weirdly enough, they checked back in 2013 when I was doing 2U, but never in my later years.

BPunjabi · « **Reply #11 on:** October 06, 2016, 11:48:05 pm »

Quote from: RuiAce on October 06, 2016, 11:39:33 pm

Weirdly enough, they checked back in 2013 when I was doing 2U, but never in my later years.

Brah I meant for HSC...

RuiAce · « **Reply #12 on:** October 06, 2016, 11:50:48 pm »

Quote from: BPunjabi on October 06, 2016, 11:48:05 pm

Brah I meant for HSC...

I know?

We're talking about the final exam and the supervisors aren't we?

BPunjabi · « **Reply #13 on:** October 06, 2016, 11:52:09 pm »

Quote from: RuiAce on October 06, 2016, 11:50:48 pm

I know?

We're talking about the final exam and the supervisors aren't we?

Wwwa...wait you did the hsc 3 years in a row?

RuiAce · « **Reply #14 on:** October 06, 2016, 11:59:03 pm »

Quote from: BPunjabi on October 06, 2016, 11:52:09 pm

Wwwa...wait you did the hsc 3 years in a row?

I was accelerated.

I did 2U in Yr 10
3U in Yr 11
And then all the rest in Yr 12

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