Im lost..
Fundamental Theorem of Calculus.
Why does y have anything to do with g?
Image coming...

- anonymous

Im lost..
Fundamental Theorem of Calculus.
Why does y have anything to do with g?
Image coming...

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- jamiebookeater

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- anonymous

Fundamental Theorem of Calculus

##### 1 Attachment

- UsukiDoll

been a while since I last explained it
and there's no good way of putting this in latex so bear with me for a bit

- anonymous

cool, thnx, I got all night :)

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## More answers

- anonymous

congrats on the big green 90 :)

- UsukiDoll

The Fundamental Theorem of Calculus
Let f be continuous on [a,b] . If F is any anti-derivative for f on [a,b], then
|dw:1437913083460:dw|

- UsukiDoll

because I can't remember the integral with the a b in latex -_-

- anonymous

oh..so g[t] in this situation is the topmost value of g[x] here.

- UsukiDoll

yes... so we need to integrate using the fundamental theorem of calculus

- anonymous

so ah..
G[a] - G[t] ?

- UsukiDoll

|dw:1437913289210:dw|
though the integration is a bit different
... it's similar to substituting
(sorry it's after 2 am in the morning so I'm a bit burned)

- UsukiDoll

that's backwards

- anonymous

oh right on
b - a oops

- UsukiDoll

|dw:1437913376771:dw|

- anonymous

t - a

- anonymous

and F is the antiderviative ?

- UsukiDoll

you're NOT taking antiderivative of g[x] dx .. for the fundamental theorem of calculus .. you are just substituting.. so ...
plug in that t inside g[x]
(again after hours... I sound off)

- UsukiDoll

"t-a"
should be
g[t]-g[a]

- UsukiDoll

your heading in your file sounds messed up... we're not using odes XD

- UsukiDoll

-_- oh I see.. at the second part.. there's an initial condition attached. -_-
when y[a]=0

- anonymous

gotcha..
so
if G is the antiderivative of g[x]
then the integration of g[x] is G[t]-G[a]
and somehow that is going to relate to y'[t] ..
Im not sure why, but I'm guessing that
possibly because there is something about all
derivatives of this kind being equivalent?

- UsukiDoll

I'm not sure. At this late at night I can't find the connection x.x but thanks for the congrats messages x.x

- UsukiDoll

y'[t] = g[t]
... feels odeish but can't connect....shutting...down..........zzzzzzzzzzzz

- Astrophysics

This is just fundamental theorem of calculus part 1

- anonymous

So indefinite integration gives you an antiderivative.
But definite integration requires an indefinite integration to get an antiderivative... and then we use that to find the definite integration as F[b]-F[a]

- anonymous

well if it didn't ask me to connect a function y to g.. and I could see how they connect, I think I would be able to answer this. I dont think I can just copy the definition of the fundamental theorem part 1, and expect to walk away with an acceptable grade

- anonymous

Ok..

- anonymous

So I'm thinking
If F is the antiderivative of f
then
f is a derivative of F
therefore if F gives us the function of a curve
then f is the rate of change on that curve.
The sum of the rate of change on that curve.. gives us the total change on the curve
Then we can get all kinds of other results, like the average.. as (sum of change)/(interval width)

- anonymous

So the change over the curve... is where the curve ends - where the curve starts

- anonymous

F[b] - F[a]
end - start

- anonymous

or the sum of all the parts.. \[\int\limits_{a}^{b} f[x] dx\]

- anonymous

of change that is.

- Astrophysics

Ok I think I see what you're doing, all you're saying is |dw:1437914782360:dw|

- anonymous

Yes on the top part.
ahh .. yes in that I am seeing f as being some kind of derivative to ... something..

- Astrophysics

Yes, in that case, that is the fundamental theorem of calculus :)

- anonymous

yes.. I see it as F'

- Astrophysics

Yes, I'm just saying F is an antiderivative of f and if we take the derivative of the antiderivative we get the derivative which is f, that was a lot..

- anonymous

so I could just write the equation as F[t]-F[a] as y[t] - y[a] and then it makes sense for y'[t] =g[t]

- anonymous

and y[a]=0 is probably irrelevant ?

- anonymous

hmm, I wonder why this rule would be sensitive to the sign of the interval.. and I wonder if they need me to explain that too.

- anonymous

I feel like a bloody bird.. this crap makes perfect sense for about 4 minutes, and then I lose it all again.

- Astrophysics

I'm just seeing what you're trying to say so, |dw:1437915317312:dw| that seems pretty good to me

- anonymous

yes, thats what Im thinking

- Astrophysics

I think that works out

- anonymous

awesome ... thanks much.. that makes sense to me too now

- Astrophysics

Haha yeah, and I'm sure others will drop by and check it out as well, but I think that looks good...

- UsukiDoll

:/ so I'm medal less in this post? seems legit. -_-

- Astrophysics

But your 90!

- anonymous

They should redo this thing medals are bitcoin.

- Astrophysics

That would be pretty awesome

- anonymous

Id give you both bitcoin

- anonymous

Ah I think that y[a]=0 clause, must be related to the idea that if y[a] = 0 then there there is nothing to subtract..

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