WEBVTT
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Okay, So our goal here is to find the
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equation of the tangent line of f of X passing
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through the point X equals a So here f of
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X is one of her ex square. And we
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want the equation. The tangent line passing through the
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point X equals one. So what do we need
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? So we're looking for the equation of a line
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. We need the slope in the slope of the
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line. It's just f prime of what? Okay
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, so the slope of the tangent line is just
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the derivative and X equals one, so we need
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to figure out what that is, first of all
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, and then we need the point on the line
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or any point on the line to write down the
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equation of the line. But we know what that
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ISS. We're just going to use the 0.1 comment
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ethic. One. We're going to use the point
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of tange INSEE. So we're going to use the
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the point where the tangent line just barely touches.
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Um, the graph of the function and this is
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actually just one what right, Because about deployment one
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, I get one. So the real question is
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what is this. Look, what is the derivative
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? That's the interesting question here. So let's just
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write down what the terrific Davis or what the slope
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of the tangent line is. So it's still limit
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. His H goes to zero of well, one
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over one plus h squared minus one over one,
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squared all over H. Okay, so just in
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case you're a little bit confused where this came from
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, remember that the derivative F Prime Obey is the
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limit is H goes to zero of f of A
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plus H minus f of a all over H.
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And so all I did here is just evaluate.
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So a is one. I just evaluated f at
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one plus h and then minus f evaluated at one
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and then divided by h. So this limit as
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well most of the time be the case when finding
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a derivative via the limit, it's going to be
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an indeterminant form. If I just try to plug
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in H equals zero, I'm going to get 0/0
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. So I'm always going to have to do more
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work, and normally that's going to involve just,
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um, typical algebraic tricks like we've seen before.
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So what I can do here is well, get
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a common denominator. So this is one plus age
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squared. This is just one. So I can
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write. This is one minus one plus h squared
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over one plus h squared all over h. So
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all I did this is just one I just wrote
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. This is one plus a squared, divided by
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one plus eight square. And then because I have
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a common denominator, I could just do the subtraction
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in the numerator. Okay, So what do we
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have here? Well, this one plus age squared
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I can actually expand that out is one plus to
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H plus H Square. And now I'm taking one
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and subtracting all of these values. So let's just
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do that. So one minus one is zero,
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and then one and then Okay, so one minus
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10 and then we're subjecting to age. So we're
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left with minus two age and then minus h squared
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. So this is a common mistake. So the
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temptation is to say, Okay, one minus 10
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and then just to h plus age squared. But
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you got to remember that you're subtracting this entire quantity
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. So really, the negative is distributed to all
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of those terms and then divided by one plus h
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squared all over h. Okay. And I'll skip
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a little bit here. So the numerator of the
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numerator has a factor of age, as does the
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denominator. So I can actually cancel one of those
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factors of age. So I'm really just left with
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the limit. His H goes to zero of minus
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two, minus h over one plus h squared.
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Okay. And now I could just evaluate. I
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just plug in h equals zero, Take the limit
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. So this is going to zero that's gone.
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And then that h is gone. One squared is
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just one, so it looks like the slope is
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just negative to Okay, so the slope is negative
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two. Then I just use my points look for
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So it looks like I have Why equals negative,
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too. Times X minus one plus one. Or
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I think that ends up being negative. Two x
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plus three. We'll just leave it like that.
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But now, because that is the equation of the
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tangent line and you see, the most important part
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is finding the derivative. Okay, So negative to
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we'll see a picture in just a second is like
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the slope of that function at X equals one.
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And of course, I'll just writing negative at two
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X plus three if you prefer soap intercept form.
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So let's look at a picture real quick. All
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right, so here is the fruit of our labor
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. We started with this function one over X Square
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, and then we were looking specifically at X equals
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one. Here's our point of tangent C 11 And
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so we know the tangent line is just gonna barely
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pass through that point. But we needed to find
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the slip, right. So then we determined that
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f prime of one was negative two. So that
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gave us the slope of the tangent line. Then
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we just use point slope form to find the equation
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of the danger line. Negative two x plus three
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. And the cool thing to do is to use
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some sort of computer algebra system into plot the line
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and be amazed when Yeah, actually, this line
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is tangent to the function. It's just barely touching
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the function. And at one point it's giving you
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You see that if I look at the blue function
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, the slope of that function is deep is well
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, it's changing, so it's starting really steep.
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And then it's kind of leveling off. And at
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X equals one. The slope of the function is
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equal to the slope of the tangent line, which
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is negative. Two. So it's telling us how
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the function is changing at the point X equals one
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. Okay, so let's do it again. We
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have so much fun last time. So what do
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we need? We need the slope, which is
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going to be f prime of zero. So we'll
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figure out what that is. That's gonna be most
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of the work. And then we need a point
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on the line to get the equation of the tangent
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line. And we're just going to use that point
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of Tange Enciso the point where the tangent line touches
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the graph and that's gonna be zero eth zero,
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which is going to be zero f of zero is
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just one. If I plug in zero for X
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, I get one. So there we go.
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So let's find the slope. Let's find the derivative
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So f prime of zero is just going to be
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square root of H plus one minus one over age
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, okay? And so why is that? Well
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, I plug in a plus H for X,
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but a zero. So I plug in zero plus
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h. So this is f of zero plus h
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. Just write that up here. This is f
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of zero plus age, and then this is f
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of zero. But we already found the f of
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zero is one. Okay, so then I just
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subtract those two quantities and divide by age and again
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, if I try to plug in H equals zero
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, I'm going to get an indeterminant form. So
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I need to do more work. And if you
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recall with limits, we have a red flag here
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we have a square root. Let's multiply by the
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kanji And let me actually not forget the most important
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part here. That really I need to take the
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limit is h goes to zero. That's important.
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Okay, so if I try to take this limit
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like I said, I'm going to get an indeterminate
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for him. So I'm gonna multiply by the kanji
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it so I'm gonna multiply by squared of H plus
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one plus one on the top and bottom and in
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the top. That's going to give me a difference
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of squares. So that's gonna just gonna be H
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plus one minus one all over H times squared H
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plus one plus one. And so again, this
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is just squared of H plus one square, which
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is H plus one minus one one square, which
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is mine. Alrighty. And so now notice that
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the numerator. So this is one minus one.
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So I'm just going to be left with H.
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But then that means thes h is will cancel.
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And, of course, again, in the limit
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is h goes to zero. I can ignore h
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equals zero. And so what I'm left with,
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it's the limit is h goes to zero of one
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over squared of H plus one plus one. But
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now I can actually take the limit. So is
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h goes to zero. This becomes one, so
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1/1 plus one is one over to. So the
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slope of the tangent line at X equals zero is
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one half already. But we have the y intercept
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. So actually, the equation the tangent line,
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it's easy to write down. It's just one half
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X plus one because the Y intercept is one when
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x zero, Why is one So this is the
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equation of the danger line. And again, this
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is, you know, a good bit of work
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to take the limit to find the derivative. But
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the cool thing about this is you get to see
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it. You get to actually see what you're finding
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. You're not just doing math Bob Loblaw doing math
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just to do it. You're doing math and then
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you get a plot it and you get to see
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Oh, my goodness, I'm actually finding something geometrically
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that's really cool. And you're going to see it
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has so many applications. So let's look at a
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picture. All right, so here it is.
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In blue, we have the function square root of
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X Plus one and we were trying to find the
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tangent line at the point X equals zero. So
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that was 01 on the graph of our function,
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and we found that f prime of zero was one
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half. So the slope of the function in blue
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at X equals zero. It's leveling off. So
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it's starting really steep the slope, and then it's
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kind of leveling off, and right here it's one
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half so it's really cool to find these slopes.
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Find these tangent lines. Look at what's going on
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. We're really developing these tools to study how functions
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are changing.