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MathEd-PII

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volunteer
which means that you know how the function is changing for any point in for any point X
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inside the function
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volunteer
So you could predictively model
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volunteer
you would like just with the derivative function, you would know exactly how the function is changing at any point in time.
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volunteer
Yeah, that's, that's the all real numbers are.
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volunteer
So this is, so this is just to prove that like, you know, it's
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volunteer
an effective
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volunteer
inaccurate way to predict the behavior of F.
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the function F, that this derivative, this derivative function.
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volunteer
Um, all that to say is that, you know, this is what we did the derivative. Now we want to figure out what is an effective and accurate way of predicting, of knowing the behavior
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volunteer
of the area under the curve, of being able to find the value under the curve.
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volunteer
which you'll be able to tell us because, you know, because it's the antiderivative, this is saying that like, OK, we have this function F.
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volunteer
uh, you get the
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volunteer
if you go back up to the
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volunteer
the first graph I did, you say we have this, we have this graph F.
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volunteer
OK. We want to know, assuming this is like, let's just say this is F.
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volunteer
right? This is the derivative function
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volunteer
and we want to know if, since this is the, this, this tells us the change in the behavior of F. We want to know, is it possible to
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volunteer
take the, like, take the and derivative, do the opposite derivative, and using the change in the behavior to be able to find out what F is.
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volunteer
But we in the way we do that is we kind of take several approaches and which are just approximations and we try to see
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volunteer
uh by taking the limit or by summing up all the infinitesimal values. Is this an accurate representation
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volunteer
of the function
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volunteer
And we know that for the derivative, any value, any Y value,
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volunteer
in the derivative tells you the slope at any point in, in the origin the parent function F.
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volunteer
But when we do that, we can say that the change, you go to Delta X and X
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XI and Xi + 1.
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volunteer
using the trapezoid rule
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volunteer
and we say
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volunteer
if we find the average
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volunteer
of the two, let's say, um, let's see.
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volunteer
Yeah, they say average
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volunteer
F, right, over some range delta X.
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volunteer
right? You see how this makes little.
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volunteer
Yeah, we can say that this is an approximation
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to
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the uh
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change in F.
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volunteer
right?
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volunteer
Because we know that um
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volunteer
since the F is the slope.
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volunteer
Let me try to, let me get some space. Let me move that graph a little lower.
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volunteer
Oh, I can't grab everything
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volunteer
I
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volunteer
very frustrating
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volunteer
Mm
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volunteer
All right
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move this
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down so I can
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volunteer
remove all of this.
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up.
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volunteer
Right
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volunteer
there, close enough
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volunteer
right
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volunteer
And you got room in the bottom
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volunteer
All right. So we know
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F
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F of X, any point is just the slope M at any point X
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volunteer
All right, just a slope
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volunteer
And we know that the slope is just equal to our change in F, right? The change and the elevation.
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volunteer
divided by the change in our X
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volunteer
And we want to know, is this area under the curve? Can we approximate the area under the curve cause we want to know what Delta F is.
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volunteer
because that's, that's our change in the function.
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volunteer
right?
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volunteer
So we do, so using the trapezoid rule, you're right, trapezoid.
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volunteer
Trapezoidal rule
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volunteer
right
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volunteer
So we can say um using our average F, right? Over the, over the step size of Delta F.
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volunteer
We say that Delta F is approximately equal to our
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average step size, which this one would be F of X_I, X of XI.
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plus F of XI + 1.
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over 2
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times our step
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right
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is approximately equal to Delta F.
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volunteer
which is the area under the curve.
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volunteer
like the entire, the full area under the curve from X1 to Xn.
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volunteer
is what our integral would be.
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volunteer
But you notice that this is missing
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volunteer
Let me use a different color
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volunteer
Um, like green, you know, that this is missing.
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this gap right here
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volunteer
So this is our
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our error value
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volunteer
right? And that's why there's Delta F remains an approximation.
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volunteer
But again, if we um
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if we take the limit
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of delta X as it approaches 0, right?
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of this delta F function.
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right?
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Then we can reasonably get
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hopefully
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volunteer
this, the accurate del delta F.
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volunteer
right? Now you notice in the equation it's multiplied, so it'd probably be equal to 0.
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volunteer
but we would, uh
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figure out, try to figure out, I guess it'd be
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volunteer
the minute it goes
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volunteer
as the error function goes to the, we'd be there, evaluate the error function.
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volunteer
the secondary error function.
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volunteer
We would want, so we, as a limit of De tax budget there are stepsize gets infinitesimally small. We want our error function.
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to be zero
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