anonymous
  • anonymous
Im lost.. Fundamental Theorem of Calculus. Why does y have anything to do with g? Image coming...
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Fundamental Theorem of Calculus
UsukiDoll
  • 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
  • anonymous
cool, thnx, I got all night :)

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anonymous
  • anonymous
congrats on the big green 90 :)
UsukiDoll
  • 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
  • UsukiDoll
because I can't remember the integral with the a b in latex -_-
anonymous
  • anonymous
oh..so g[t] in this situation is the topmost value of g[x] here.
UsukiDoll
  • UsukiDoll
yes... so we need to integrate using the fundamental theorem of calculus
anonymous
  • anonymous
so ah.. G[a] - G[t] ?
UsukiDoll
  • 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
  • UsukiDoll
that's backwards
anonymous
  • anonymous
oh right on b - a oops
UsukiDoll
  • UsukiDoll
|dw:1437913376771:dw|
anonymous
  • anonymous
t - a
anonymous
  • anonymous
and F is the antiderviative ?
UsukiDoll
  • 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
  • UsukiDoll
"t-a" should be g[t]-g[a]
UsukiDoll
  • UsukiDoll
your heading in your file sounds messed up... we're not using odes XD
UsukiDoll
  • UsukiDoll
-_- oh I see.. at the second part.. there's an initial condition attached. -_- when y[a]=0
anonymous
  • 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
  • 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
  • UsukiDoll
y'[t] = g[t] ... feels odeish but can't connect....shutting...down..........zzzzzzzzzzzz
Astrophysics
  • Astrophysics
This is just fundamental theorem of calculus part 1
anonymous
  • 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
  • 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
  • anonymous
Ok..
anonymous
  • 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
  • anonymous
So the change over the curve... is where the curve ends - where the curve starts
anonymous
  • anonymous
F[b] - F[a] end - start
anonymous
  • anonymous
or the sum of all the parts.. \[\int\limits_{a}^{b} f[x] dx\]
anonymous
  • anonymous
of change that is.
Astrophysics
  • Astrophysics
Ok I think I see what you're doing, all you're saying is |dw:1437914782360:dw|
anonymous
  • anonymous
Yes on the top part. ahh .. yes in that I am seeing f as being some kind of derivative to ... something..
Astrophysics
  • Astrophysics
Yes, in that case, that is the fundamental theorem of calculus :)
anonymous
  • anonymous
yes.. I see it as F'
Astrophysics
  • 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
  • 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
  • anonymous
and y[a]=0 is probably irrelevant ?
anonymous
  • 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
  • anonymous
I feel like a bloody bird.. this crap makes perfect sense for about 4 minutes, and then I lose it all again.
Astrophysics
  • Astrophysics
I'm just seeing what you're trying to say so, |dw:1437915317312:dw| that seems pretty good to me
anonymous
  • anonymous
yes, thats what Im thinking
Astrophysics
  • Astrophysics
I think that works out
anonymous
  • anonymous
awesome ... thanks much.. that makes sense to me too now
Astrophysics
  • Astrophysics
Haha yeah, and I'm sure others will drop by and check it out as well, but I think that looks good...
UsukiDoll
  • UsukiDoll
:/ so I'm medal less in this post? seems legit. -_-
Astrophysics
  • Astrophysics
But your 90!
anonymous
  • anonymous
They should redo this thing medals are bitcoin.
Astrophysics
  • Astrophysics
That would be pretty awesome
anonymous
  • anonymous
Id give you both bitcoin
anonymous
  • 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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