WEBVTT
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So now that we've finished our topic on limits,
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we've built up all of the machinery that we need
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to go back to. The question of finding slopes
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of functions are slopes of the graphs of functions or
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, in other words, finding instantaneous rates of change
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. And now another word for this that will use
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from here on out is the derivative. And put
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simply, the derivative really is just the instantaneous rate
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of change at the function or the slope of the
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tangent line at a point. So let's just draw
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a picture just to remind you of the set up
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. So we have a function that I'll sketch.
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Maybe it looks something like that. And so if
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I have a point, X equals a then what
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I really am looking for is this tangent line here
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. Now there's a question about whether or not this
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tangent line exists that will sort of discuss. But
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if it exists, this tangent line will come and
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touch the graph of the function, and exactly one
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point so specifically it will touch the function at this
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point. A. That's a bit so. The
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derivative is nothing more than the slope of this line
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. So this line is gonna have an equation y
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equals mx plus B. And by the derivative,
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we really just mean the slope. Yeah, And
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now that we again, we have this machinery of
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limits to be able to take the limit of secret
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lines and tangents really are just the limit of secret
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lines. And that's what we're going to be doing
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in this topic. So the first thing we're going
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to dio is we're just going to define the derivative
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. We're going to very precisely say what the derivative
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is in terms of a limit. Now that we
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understand limits, weaken, define the derivative, then
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we're just going to practice finding derivatives. And specifically
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, what I mean is we're going to be finding
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what's called derivative functions. So we're gonna have a
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function, and then we're going to be able to
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find another function that the outputs of that function are
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actually the derivatives, the slope of the function itself
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or specifically, the slope of the tangent minds.
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And we're going to do this with all different types
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of functions. Polynomial is rational functions, Trina Metric
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functions, exponential functions, etcetera. We're just going
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to cover everything and then once we do that,
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we're going to talk about some applications and specifically what
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I mean by applications here is applications of the derivative
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or the instantaneous rate of change of a function at
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a point. And now these will span. All
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different areas of science will probably bend a little bit
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more towards the physical applications because they're very concrete,
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but will also touch on a few other types of
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problems that occur in economics and biology. Stuff like
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that. So you're going to see we're going to
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define the derivative, and you're going to see that
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it's an extremely useful and applicable concept, really across
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the board, in science and just in the way
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that our world works. And again, it's going
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back to this key idea about calculus. The calculus
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really is about the study of change of a function
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, and the derivative is giving us a that first
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tool of saying, okay, this is how a
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function is changing point by point