WEBVTT
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Okay, So before we define the derivative, let's
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go back to one of the first concepts we discussed
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in the last topic. And that was the average
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rate of change of a function over an interval.
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So recall that the average rate of change of the
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function ffx over some interval. And now I'm going
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to write the interval in a little bit different way
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. I'm not just going to write a to B
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. I'm actually going to write a thio A plus
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h where h here. I'm just thinking about is
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being a positive number. Okay, so I'm just
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kind of parameter rising the interval by its length.
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Okay, So h is the length of the interval
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and of course, the interval ISS starting at.
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And if I just let h equal b minus a
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which is the length of the interval. You see
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that this really is just a to B, Okay
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, so don't think too much about that. But
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over this interval, the average rate of change of
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the function is just f of April's H minus half
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a day all over h okay. And notice again
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that I'm dividing by the length of the interval which
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is B minus a, but I'm saying B minus
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a may be able to put that down. So
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be mind to say I'm thinking about is h based
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on how we defined average rate of change before and
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so just a note. This is the slope of
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the secret line between two points on the graph of
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dysfunction. So between the points A and that's a
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day and then a plus age, an F of
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April's h okay. And of course we have this
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nice picture of the Secret Line. So we had
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the function of X and then the secret line.
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So this is a and over here is April's age
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than the Secret Line just connects these two points.
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Okay, But of course, we're also interested in
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the instantaneous rate of change, and in fact,
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that's going to be a little bit more useful for
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our applications. So we're really looking for the tangent
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line at X equals that So the average rate of
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change is over and honorable. So here's our interval
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. The instantaneous rate of change or what we'll call
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the derivative is going to be at a point,
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and so what we're going to do is we're really
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gonna let this h good zero and watch these secret
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lines level off to the tangent Mark, assuming that
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our function has a derivative at that point. Okay
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, so now that we've sort of reviewed and I
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keep saying this, we're derivative. Let me actually
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tell you specifically what the derivative iss. All right
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. So the derivative of a function at X equals
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a So the notice the difference between average rate of
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change and what we're defining here, the average rate
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of change is over an interval, and the derivative
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is at a specific point. It's an instantaneous rate
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of change. So the derivative at X equals a
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is given by, Well, it's basically this same
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difference question that we had for average rate of change
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. But again, we won't let that h go
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to zero. And the whole reason we went through
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this process of talking about limits is to have a
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way to precisely say H is getting closer and closer
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to zero. So we're taking the limit is h
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goes to zero, and now we introduce some notation
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. That's what this kind of colon equal sign means
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. So I'm going to define this notation to mean
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the derivative instead of saying f obey. Well,
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that's the function value. I'm going to say f
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prime of a. So this is the derivative right
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here. So when you see f prime of a
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all it is is it's the limit of this difference
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question. So you're just taking those seeking lines in
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their slope the average rate of change squeezing them in
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around some point X equals a and that's giving you
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the derivative. Now, one important note is that
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this definition Onley makes sense. If this limit exists
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so very important kind of caveat is that this is
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only defined if the limit exists. And for now
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, we'll just assume that this limit exists when we
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talk about its properties. So here we go,
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the derivative What is the driven? Okay, so
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I sort of just made up a vocab word derivative
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. You know, it doesn't mean anything if I
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just say it, I need to tell you what
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it really is. The derivative is the instantaneous rate
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of change as well as the slope of the tangent
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line. Okay, So, again, if you
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want to go back and think about our velocity example
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so the average rate of change was the ridge velocity
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of a position function over a time interval. But
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if you look on the speedometer in your car,
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that tells you the instantaneous rate of change. So
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actually, the derivative is that instantaneous speed that you're
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traveling or instantaneous velocity. If you are thinking about
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well, I'm not going forward or backward, but
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we'll talk about that later when we talk about applications
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. But this is what you need to know.
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For now, the derivative it has, this technical
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definition is a limit. That's how you actually compute
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it. You if you want to figure out what
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it is, so if it's two or five or
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negative three or whatever it is, that's the slope
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of that tangent line geometrically. But its interpretation is
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that it's the instantaneous rate of change of the function
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. It's the slope, if you will, the
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function, which of course, is changing. So
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let me give you a quick sketch. So again
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, if we have a function and we'll just kind
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of sketch it like this, you get sort of
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arbitrary, and then we have a point. X
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equals a somewhere right here. Then our tangent line
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, assuming it exists. Of course, assuming that
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limit exists, that limit exists. Then the tangent
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line exists. Okay, so here's ffx or function
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, and then we have a tangent line and we'll
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be a little bit more specific about what the tangent
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line is in a second. But the derivative so
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f prime of a is equal to this number.
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Yeah, the slope of the tangent line. Okay
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, so geometrically. It's very clear what the derivative
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is telling us. And actually, if we want
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the equation of the tangent line, notice that for
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the tangent line we have the slope. And if
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we want a point on the line, well,
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we know a point on the line. It's a
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common F obey member that this point is the point
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of Tange insee for the tangent line. It's the
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point where the the tangent line just barely touches the
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function. So a f obey is a point on
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the line. So the tangent line really is why
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, minus f of s I'm just using point slope
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form equals m times X minus A or if you
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prefer, I can add s obey over. So
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this is kind of the general form for the tangent
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line right there. That's how you find it.
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So in order to find the tangent line, you
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need the slope. By taking that limit. Finding
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the derivative at X equals a and then just using
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this point of tange Insee, a common method to
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find the equation of the tangent line. So the
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next thing we need to do is sort of go
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back to this question. Does the derivative always exist
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? And of course, the answer is going to
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be No, because it's defined is a limit.
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And as we've already seen in our discussion on limits
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, limits don't always exist. So the derivative F
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prime of a does not always exist, but just
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to introduce um, or notation or some more terminology
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if the derivative exists. So if f prime of
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a exists or, in other words, if the
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tangent line exists of the function, or we'll see
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later if the function has what's called a linear approximation
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at that point. So if F prime of exists
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, then F of X is what's called Differential at
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X equals a and now this is going to be
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more related to when we talk about derivative functions.
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But for now, we just want to introduce this
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terminology because it'll just kind of get thrown out.
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It's just in our mathematical language. The derivative exist
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a functions differential. The tangent line exists at that
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point, although there's just a lot of different ways
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to say the same thing. And probably that's why
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math is confusing. That's why anything is confusing.
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If there was only one way to say something,
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then it would probably be easier toe learn. But
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it wouldn't be as broad and kind of rich in
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content because there's all these different ways to think about
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the derivative or think about differential bility. But the
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key thing when you're learning this for the first time
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is to realize that all of these concepts are exactly
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the same thing. The instantaneous rate of change,
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the slope of the tangent line, the derivative differential
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ability. They're all just saying the same thing.
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So let's get into the really interesting question of talking
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about when a function is not differential. We're gonna
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spline spend plenty of time talking about functions that do
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have derivatives. That's going to be an essential part
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of this class, but we're defining this concept.
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And so it's gonna be worth while to think about
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cases where the derivative does not exist. Okay,
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so when is a function not differential will recall that
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the derivative is defined as a limit. So this
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question when is epics not differential is the same as
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this question. When does f prime of a not
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exist? And when I say not exist, I
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mean as a limit because this is a limit.
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When ordered to answer this question, we got to
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think back. So how can a limit not exist
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? Well, there's some really, really complicated cases
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that we're not going to consider. We're really just
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going to consider the simplest case. Okay, when
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As a limit, the left hand limit doesn't equal
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the right hand. So we're gonna introduce some work
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terminology, but it's really nothing new. So we're
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going to define what's called the left hand derivative.
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And if you're really tracking, you're going to know
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exactly what this is. It's just the left hand
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limit of F prime event. So instead of just
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taking the limit is H goes to zero. We're
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just going to take the limit is H goes to
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zero from the left of this difference question and then
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similarly, will define the right hand derivative. So
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this is really nothing more than looking at the slope
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from the left and from the right. What is
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the slope approaching? If they're approaching the same thing
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, then this limit will exist and the function will
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be differential. Okay, so we need these two
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things to be equal. That's the key point for
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ethics to be differential. Well, the left hand
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drive it. It has to equal the right hand
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derivative. And in the first example, we'll see
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if a function that's not differential, that's exactly what
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we'll do. We'll look at the left hand derivative
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, and I look at the right hand directive and
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show that they're not equal. Therefore, the function
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will not be differential. Okay, so let's look
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at everybody's favorite function now. Of course, I'm
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being sarcastic. I doubt this is everybody's favorite function
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, but the function we're going to look at is
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absolute value of X, and I'm just going to
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explicitly write down absolute value of X as a piece
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wise function to be very clear. What it iss
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so recall. The absolute value of X is just
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going to be X when X is greater than or
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equal to zero. But it's actually going to be
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minus X when X is less than zero. Okay
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, so here's the question. What is F prime
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of zero? What's the derivative of the function at
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zero? Or, in other words, what's the
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slope of the tangent line passing through zero? Well
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, let's just sketch what this function looks like,
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first of all, just to kind of illustrate what
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the problem is going to be. So this function
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looks like this. It looks like Y equals eggs
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going this way, but it looks like why equals
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negative X going this way. So if you look
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at all of these points, so like if you
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look at negative one, then the slope is just
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going to be the slope of this line. Negative
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one. Or if I look on the other side
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like it one, the slope or the derivative is
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just going to be one right, because these air
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just lines. They're just kind of put together in
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pieces. But what about zero? What is this
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slope of this function at zero. Well, let's
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look at the two limits. Let's look at the
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left hand derivative in the right hand derivative. So
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the limit is h approaches zero from the left of
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absolute value of zero plus H minus absolute value of
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zero all over. H. Well, this is
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just the limit is h goes to zero from the
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left of absolute value of H over H. Okay
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, but kind of saving, You know, all
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of the steps. This is something we saw with
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limits. This is a one sided limit and because
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H is approaching zero from the left absolute value of
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age, we want to use this definition. So
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this is really minus age over age or minus one
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. And that really makes a lot of sense.
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Because if you just think about the slope is as
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you purchase your from the left, it's negative one
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, right? It's just a slope. It's just
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a line passing through the origin of slope negative went
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and then the right handed derivative. It's going to
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be the same thing. Except this time when we
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look at absolute value of age were to the right
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of zero. So we're just going to use H
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. So this is really gonna be H over H
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, which is one. So the limit is just
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one. And again, that makes sense, because
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to the right, our function is just y equals
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X, which has slope one. So here we
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say the absolute value of X is not differential at
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Syria, but notice that actually f is continuous and
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zero, and this is something that will mention later
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. It actually being differential is stronger than being continuous
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. So, in other words, if you're differential
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than you will be continuous at that point in another
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way, to think about different ability is what we'll
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talk about next. The different ability is really talking
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about smoothness in some sense. So again, let's
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look at just kind of the geometric meaning of differential
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bility. So if a function is differential, then
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F is sometimes said to be smell at X equals
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day. And this really goes back to the idea
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of looking at how function is changing. If a
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function is differential, then somehow it's changing nicely there
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. It's just kind of accelerating or decelerating. In
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some sense, there's not any sort of jerking going
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on, and so let's just look at a few
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examples of when a function is not differential and see
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00:24:42.009 --> 00:24:45.960 A:middle L:90%
the types of situations where a function cannot be smooth
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in some sense. So let's look at some examples
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. So one example we've already seen. That's when
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a function has what's called a corner. And that's
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the case, like for absolute value of X.
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00:25:26.440 --> 00:25:30.069 A:middle L:90%
And so you see that a corner is not smooth
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, right? So if you kind of trace your
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hand along the corner, it doesn't roll Canada from
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one. You know, it doesn't change and roll
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around the corner. It's sharp as a sharp corner
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00:25:47.700 --> 00:25:49.269 A:middle L:90%
, so it's not smooth. Okay, so sharp
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means not smooth. Okay? And again, we
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00:25:53.910 --> 00:25:57.259 A:middle L:90%
saw this by showing that the left hand derivative doesn't
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00:25:57.259 --> 00:26:00.849 A:middle L:90%
equal the right hand derivative. So if we kind
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00:26:00.849 --> 00:26:03.049 A:middle L:90%
of approached the corner from the left, we have
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00:26:03.049 --> 00:26:04.000 A:middle L:90%
a slope going like this. Like if you look
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00:26:04.000 --> 00:26:07.430 A:middle L:90%
at the top of the house like if I go
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up one side, I have a slope this way
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00:26:08.420 --> 00:26:11.059 A:middle L:90%
and I go up the other way. I have
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00:26:11.059 --> 00:26:14.089 A:middle L:90%
a slope this way, and they needed a corner
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00:26:14.150 --> 00:26:17.660 A:middle L:90%
like that. Then it's not Smith. Okay,
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00:26:18.940 --> 00:26:23.380 A:middle L:90%
so another example is what's called a cusp in a
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00:26:23.380 --> 00:26:26.420 A:middle L:90%
cusp more or less is the same thing as the
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corner but a cusp instead of just kind of meeting
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at two different angles. Cusp really is, um
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00:26:41.140 --> 00:26:49.589 A:middle L:90%
, coming down in smoothing out and then coming back
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the other way. So really, the left and
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00:26:55.380 --> 00:27:00.680 A:middle L:90%
right hand derivatives air becoming undefined so the limits aren't
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existing. But somehow there they're infinitely. So 11
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00:27:04.630 --> 00:27:08.130 A:middle L:90%
is approaching infinity and one's approaching minus infinity. Not
322
00:27:08.130 --> 00:27:12.670 A:middle L:90%
just being left and right. So, you know
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00:27:12.670 --> 00:27:15.259 A:middle L:90%
, from the left here, you get a number
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00:27:15.259 --> 00:27:15.609 A:middle L:90%
from the right here. You get a number.
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00:27:15.609 --> 00:27:19.309 A:middle L:90%
They're not equal here. You're really getting that.
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00:27:19.309 --> 00:27:22.680 A:middle L:90%
The limit is an infinite limit, right? And
327
00:27:22.680 --> 00:27:25.910 A:middle L:90%
so from the left, the slope is approaching minus
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00:27:25.910 --> 00:27:27.400 A:middle L:90%
infinity, and from the right, the slope was
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00:27:27.400 --> 00:27:32.099 A:middle L:90%
approaching. Infinity is the idea. So this is
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00:27:32.099 --> 00:27:42.319 A:middle L:90%
really thinking about infinite slopes, whereas here and kind
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00:27:42.319 --> 00:27:45.269 A:middle L:90%
of ah limit. Since here we're just thinking about
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00:27:45.269 --> 00:27:49.799 A:middle L:90%
those limits not being equal. And then the final
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00:27:49.799 --> 00:27:52.599 A:middle L:90%
example that will look at is what's called a vertical
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00:27:52.599 --> 00:27:59.440 A:middle L:90%
tangent. And this is similar to a cust.
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00:27:59.769 --> 00:28:07.390 A:middle L:90%
Except that actually there's some consistency in the left and
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00:28:07.390 --> 00:28:10.849 A:middle L:90%
right hand limits that they're actually both either approaching infinity
337
00:28:10.849 --> 00:28:23.740 A:middle L:90%
or approaching minus infinity. And so, if our
338
00:28:23.740 --> 00:28:26.170 A:middle L:90%
function looks something like this, so we go up
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00:28:26.740 --> 00:28:30.859 A:middle L:90%
and we see that our tangent line, the slope
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00:28:30.339 --> 00:28:36.519 A:middle L:90%
is actually becoming undefined. So the tangent line exists
341
00:28:37.240 --> 00:28:44.680 A:middle L:90%
here, but the slope is undefined. So Tangent
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00:28:44.680 --> 00:28:59.240 A:middle L:90%
line has undefined slip. Or, in other words
343
00:28:59.579 --> 00:29:03.599 A:middle L:90%
, the tangent line is vertical. I mean,
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00:29:03.599 --> 00:29:07.619 A:middle L:90%
hence the name vertical tangent. So actually the Tangela
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00:29:07.619 --> 00:29:11.140 A:middle L:90%
exists, but it just it's burke. It has
346
00:29:11.140 --> 00:29:15.029 A:middle L:90%
an undefined slope. So if you look at the
347
00:29:15.029 --> 00:29:18.579 A:middle L:90%
left hand derivative in the right hand derivative in this
348
00:29:18.579 --> 00:29:21.599 A:middle L:90%
case, they would actually both be approaching infinity,
349
00:29:21.769 --> 00:29:25.480 A:middle L:90%
whereas with a cust in one direction, you're approaching
350
00:29:25.480 --> 00:29:29.259 A:middle L:90%
infinity in another direction. You're approaching minus infinity.
351
00:29:29.400 --> 00:29:33.309 A:middle L:90%
So there's really not a well defined tangent line in
352
00:29:33.309 --> 00:29:37.710 A:middle L:90%
that case, just like with a corner. So
353
00:29:37.710 --> 00:29:41.109 A:middle L:90%
you see these air just three examples of cases where
354
00:29:41.109 --> 00:29:44.950 A:middle L:90%
functions can fail to be differential, okay? Or
355
00:29:44.950 --> 00:29:48.069 A:middle L:90%
they could fail to be smooth in some sense.
356
00:29:48.079 --> 00:29:48.970 A:middle L:90%
So these air just important to keep in the back
357
00:29:48.970 --> 00:29:52.160 A:middle L:90%
of your mind and in the examples will actually see
358
00:29:52.470 --> 00:29:56.170 A:middle L:90%
specific cases where these come up