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  • - [Voiceover] What I wanna do in this video

  • is a few examples that test our intuition

  • of the derivative as a rate of change

  • or the steepness of a curve

  • or the slope of a curve

  • or the slope of a tangent line of a curve

  • depending on how you actually want to think about it.

  • So here it says F prime of five

  • so this notation, prime

  • this is another way of saying well what's the derivative

  • let's estimate the derivative of our function at five.

  • And when we say F prime of five this is the slope

  • slope of tangent line

  • tangent line at five

  • or you could view it as the

  • you could view it as the rate of change

  • of Y with respect to X

  • which is really how we define slope

  • respect to X of our function F.

  • So let's think about that a little bit.

  • We see they put the point

  • the point five comma F of five right over here

  • and so if we want to estimate the slope of the tangent line

  • if we want to estimate the steepness of this curve

  • we could try to draw a line

  • that is tangent right at that point.

  • So let me see if I can do that.

  • So if I were to draw a line starting there

  • if I just wanted to make a tangent

  • it looks like it would do something like that.

  • Right at that point that looks to be

  • about how steep that curve is

  • now what makes this an interesting thing in non-linear

  • is that it's constantly changing

  • the steepness it's very low here and it gets steeper

  • and steeper and steeper as we move to the right

  • for larger and larger X values.

  • But if we look at the point in question

  • when X is equal to five remember F prime of five

  • would be if you were estimating it

  • this would be the slope of this line here.

  • And the slope of this line it looks like

  • for every time we move one in the X direction

  • we're moving two in the Y direction.

  • Delta Y is equal to two when delta X is equal to one.

  • So our change in Y

  • with respect to X

  • at least for this tangent line here

  • which would represent our change in Y

  • with respect to X right at that point

  • is going to be equal to two over one, or two.

  • And it's almost estimated, but all of these are way off.

  • Having a negative two derivative

  • would mean that as we increase our X our Y is decreasing.

  • So if our curve looks something like this

  • we would have a slope of negative two.

  • If having slopes in this

  • a positive of point one

  • that would be very flat something down here

  • we might have a slope closer to point one.

  • Negative point one that might be closer on this side

  • now we're sloping but very close to flat.

  • A slope of zero, that would be right over here at the bottom

  • where right at that moment as we change X

  • Y is not increasing or decreasing

  • the slope of the tangent line right at that bottom point

  • would have a slope of zero.

  • So I feel really good about that response.

  • Let's do one more of these.

  • So alright, so they're telling us to compare

  • the derivative of G at four to the derivative of G at six

  • and which of these is greater

  • and like always, pause the video

  • and see if you can figure this out.

  • Well this is just an exercise

  • let's see if we were to

  • if we were to make a line that indicates the slope there

  • you can do this as a tangent line

  • let me try to do that.

  • So now that wouldn't, that doesn't do a good job

  • so right over here at

  • that looks like a

  • I think I can do a better job than that

  • no that's too shallow to see

  • not shallow's not the word, that's too flat.

  • So let me try to really

  • okay, that looks pretty good.

  • So that line that I just drew

  • seems to be indicative of

  • the rate of change of Y with respect to X

  • or the slope of that curve

  • or that line you can view it as a tangent line

  • so we could think about what its slope is going to be

  • and then if we go further down over here

  • this one is, it looks like it is

  • steeper but in the negative direction

  • so it looks like it is steeper for sure

  • but it's in the negative direction.

  • As we increase, think of it this way

  • as we increase X one here

  • it looks like we are decreasing Y by about one.

  • So it looks like G prime of four

  • G prime of four, the derivative when X is equal to four

  • is approximately, I'm estimating it

  • negative one

  • while the derivative here when we increase X

  • if we increase X by

  • if we increase X by one

  • it looks like we're decreasing Y by close to three

  • so G prime of six

  • looks like it's closer to negative three.

  • So which one of these is larger?

  • Well, this one is less negative

  • so it's going to be greater than the other one

  • and you could have done this intuitively

  • if you just look at the curve

  • this is some type of a sinusoid here

  • you have right over here the curve is flat

  • you have right at that moment

  • you have no change in Y with respect to X

  • then it starts to decrease

  • then it decreases at an even faster rate

  • then it decreases at a faster rate

  • then it starts, it's still decreasing

  • but it's decreasing at slower and slower rates

  • decreasing at slower rates and right at that moment

  • you have your slope of your tangent line is zero

  • then it starts to increase, increase, so on and so forth

  • and it just keeps happening over and over again.

  • So you can also think about this in a more intuitive way.

- [Voiceover] What I wanna do in this video

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Derivative as slope of curve | Derivatives introduction | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 12 日
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