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Okay, this is a video on the objective 1.13, objective 1.13, which is given a Cartesian
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graph of a function be able to compare instantaneous rates with rates of change with instantaneous
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rates of change with average rates of, oops, that got cut off, average rates of change.
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Okay, so for example, if I want to do an instantaneous rate of change, well that's the derivative,
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right? instantaneous rates of change are the derivatives, that's the derivative, right?
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So say I want to compare the derivative, the average instantaneous rate of change at 2 to the
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average rate of change from 1 to 3. So there's the average rate of change from 1 to 3.
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Okay, so what am I doing here in the graph? The average rate of change from minus 3 to 3 would
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be go over to minus 3, go up to the graph, make a little point right here, go over to the
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graph at 3, go down, make a little point, draw a line between them, that's that secant line,
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right? The secant line that represents the average rate of change, then as best you can,
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go down to 2 and try to draw on the tangent line, that's the red line. So the red line is
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a tangent line, the blue line is the average rate of change from minus 3 to 3, the red line is the
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instantaneous rate of change, I eat a derivative at 2. And if I was to compare these two,
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if I was to compare them, what could I say here? What could I say? Would I say that this one is
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bigger than this one? Which one is bigger? This one is actually steeper, right? But it's
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negative, right? And this one is less steep, but it's negative. So a smaller negative
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number is technically bigger than a larger negative number. So in this case, I would put
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it a greater than sign in there, okay? So don't get confused, it isn't the absolute value
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of the slope that we're comparing, it's the actual slope with the sign in there. So this
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one is actually smaller, the blue one is actually smaller than the red one because its slope is
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bigger than negative. And then let's see, how about this next one? What if I want to compare
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the average rate of change between minus 2 and 2 to the derivative, the instantaneous rate of
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change is 0. Okay, that means I got to come over here and change my grass. So here I'm
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going to go from 2 and I'm going to go from negative 2 to 2. Okay, so there's my average
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rate of change graph and now I'm going to make the derivative or the instantaneous rate of
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change happen at 0. And you can tell us something here which is unusual about the tangent at the
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derivative inflection is it's the only place where the tangent will actually cross, will
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actually cross the function. So here the tangents on the opposite side, then it is over
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here. Yet everywhere else, everywhere else I go with this tangent notice, everywhere else,
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the tangent is always on the same side of the curve right there at that point, right?
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Obviously, you know, I'm not talking about everywhere, right? But right there at the point it's
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on the same side and the only place where a tangent line can cross a curve is add
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and the reflection point right there. So now if I compare these two, what do I get?
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Well, the blue one is negative, the red slope is also negative and because a smaller
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negative number is actually bigger, then the blue one is actually bigger. So I have to
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go in there and I have to say this, say the average rate of change from minus 2 to 2
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is actually bigger than the instantaneous rate of change at 0. Why? Because a, a,
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this is a smaller slope, it's not quite as steep and it's negative and this one is
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a little steeper and negative. So this number is actually what?
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Smaller than the slope of this line, smaller than the slope of that line. Okay, well,
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there's always, there's lots of other possibilities. I could, I could go through here all day long
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and I could change my, I could change my average rate of change for example,
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I could go like that and I could come in here and make this guy go up here and then I could
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start comparing average rates of change to the instantaneous ones,
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what would be an interesting comparison? Like that one right there, what would that be?
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What would that be?
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Oh, let's stop that right there. What would that be? Well, obviously this one is
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negative, this one is positive, so the average rate of change is much, much bigger than
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the instantaneous rates of change. Of course, if I wanted to, I could probably find,
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I could probably find a rate of change where it was bigger. Okay, there.
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So there an average rate of, instantaneous rate of change is now bigger than the average rate of change.
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Right? So that's all you're doing in this one is trying to compare those two and get a clear
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idea about when an instantaneous rate of change is bigger than an average rate of change.
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So that's the average rate of change.