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MathEd-PII

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volunteer
because the definition of a tangent line means that the, the line can only intersect with a function at one point
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volunteer
and we define that point as X1.
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volunteer
X1Y1
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student
ok ok
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volunteer
Any more questions before I continue?
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volunteer
That's a very good question.
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student
nope
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volunteer
Sorry, my handwriting can get a little bad
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volunteer
Please let me know if I need to, um, kind of enlarge something.
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volunteer
or rewrite it
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volunteer
Um
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volunteer
OK. So back to our um solution for Y2.
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volunteer
OK.
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volunteer
So if we remember back to our slope equation
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volunteer
right? I didn't finish writing it, but you know, uh, if you look at the denominator
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volunteer
this X2 minus X1. Doesn't that just look like our delta X?
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student
yes
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volunteer
Yeah. So we factor this back in here.
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volunteer
right? So we find that M is equal to
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volunteer
um we'll say Y to uh
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volunteer
yeah, I will say F(X2) minus
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volunteer
F of X1 over delta X.
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volunteer
times delta X + F(X1)
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volunteer
Um, what does that give us
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volunteer
I guess it's just give us X of X1 is equal to X1, that doesn't help us.
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volunteer
But I guess we can leave this
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volunteer
leave that as is.
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volunteer
Um, cause I, I fly plug it back in. This is just going to give me that F of X1 is equal to Y1, which we already known, doesn't really help us.
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volunteer
but we could rewrite this in terms of
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volunteer
uh instead of this being Y2, we can rewrite this as M delta X + F of X1.
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volunteer
So that
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volunteer
if we
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volunteer
sorry, go ahead
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volunteer
Did you have a question
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volunteer
OK. So, if we leave it
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volunteer
of this form
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volunteer
um what Y2 is and we factor this back into our error equation.
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volunteer
right? We find that
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volunteer
uh
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volunteer
M delta X.
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volunteer
plus F of X1.
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volunteer
minus F of X2, right? And then we write it all out, we find that
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volunteer
um
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volunteer
the error function is just the, whatever our step side is, tell them to slope.
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volunteer
minus, we could say Delta F.
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volunteer
right, with the actual function is actually, let me, let me just try to solve for.
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volunteer
no F of X1
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volunteer
Yeah, we said
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volunteer
Yeah, that's fine. You can just say F of X1.
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volunteer
plus F of X1.
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volunteer
which is just equal to
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volunteer
M delta X is equal to minus.
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volunteer
RDF
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volunteer
All right, which is basically just the difference in a function is equal to whatever our change in why is, whatever our slope is.
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volunteer
Um.
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volunteer
and
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volunteer
if we can get
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volunteer
this is where we start taking, yeah, sorry.
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volunteer
Let me enlarge this.
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volunteer
Give me one second I think. I did this, I did this exact derivation
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volunteer
a couple of weeks ago. I'm think I'm trying to make sure I remember everything correctly.
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volunteer
Uh, don't forget anything.
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student
ok take your time]
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volunteer
I found a tangent line um found the air functioning, found the error value.
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volunteer
I think we just want the error value in terms of
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volunteer
why one and S's.
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volunteer
Yeah.
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volunteer
Yeah, sorry. I feel like this delta, this, this error function is wrong.
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volunteer
Somehow it's close to being around. I feel like I'm missing something.
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volunteer
Cause once you consider that er cause like the air function
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volunteer
if you take a limit
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volunteer
as delta X
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volunteer
goes to 0, right? That we make our step size infinitesimally small.
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volunteer
right? We make our X2
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volunteer
kind of really close to X1, right? You'll notice that our air function
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volunteer
should equal 0
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volunteer
right? At that, at that point, you would say that Y2 is equal to f of X2.
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volunteer
right? If your, if your air function is absolutely zero. And this is where we can um
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volunteer
say that our approach using the tangent line.
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volunteer
is an effective approach to evaluating um
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volunteer
the next values in the function, right? That it can be used effectively and if you had um
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volunteer
if you had the
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volunteer
tangent, if you had the slope
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volunteer
at any point in X. So you had M
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volunteer
for every X
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volunteer
in the domain, right? All real numbers that the function lies.
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volunteer
then you should be able to approximate
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volunteer
all the values of F of X.
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volunteer
Um, there is
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volunteer
Yeah
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volunteer
All right, I'm gonna cheat
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volunteer
Actually, I can't, I don't know if I can find that.
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volunteer
Um.
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volunteer
anyway, and if you had, and if you did the sum,
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volunteer
or if you
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volunteer
if you did the set of all slopes for every X.
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volunteer
you would have the function F of X, which would give you all the information of the change in behavior.
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volunteer
of F
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volunteer
right? So what a derivative actually is
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volunteer
is just, it tells you the behavior of the function F.
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