If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:6:10

AP.CALC:

CHA‑2 (EU)

, CHA‑2.B (LO)

, CHA‑2.B.3 (EK)

, CHA‑2.B.4 (EK)

, CHA‑2.C (LO)

, CHA‑2.C.1 (EK)

what I want to 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 5 so this notation prime this is another way of saying well what's the derivative let's estimate the derivative of our function at 5 and when we say F prime of 5 this is the slope slope of tangent line tangent line at 5 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 5 comma F of 5 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 and 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 it tangent it looks like it would do something like that that right at that point that looks to be about how steep that curve is now what makes this an interesting thing and nonlinear is that it's constantly changing 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 5 remember F prime of 5 would be if we 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 2 over 1 or 2 and they told us to estimate it but all of these are way off having a negative 2 derivative would mean that as we increase our X are Y is decreasing so if our curve looks something like this we would have a slope of negative 2 if having having slopes in this a positive of 0.1 that would be very flat something down here we might have a slope closer to point 1 negative point 1 that might be closer on this side whereas downward sloping but very close to flat a slope of 0 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 0 so I feel really good about that response let's do one more of these so all right so they're telling us to compare the derivative of G at 4 to the derivative of G at 6 and which one of these is greater and like always pause the video and see if you could 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 could view this as a tangent line so let me try to do that so that wouldn't that that's an do a good job so right over here at so that looks like a pretty I think I can do a better job than that now that's too shallow let's see not shallows number 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 at that line you could view it as a tangent line so that we can think about what its slope is going to be and 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 it is steeper for sure but it's in the negative direction as we increase think of it this way as we increase X 1 here it looks like we are decreasing Y by about 1 so it looks like G prime of 4 G prime of 4 the derivative when X is equal to 4 is approximately I'm estimating it negative 1 while the derivative here when we increase X if we increase X by if we increase X by 1 it looks like we're decreasing Y by close to 3 so G prime of 6 looks like it's closer to negative 3 so which one of these is 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 sinusoidal curve is flat it's you have right at that moment they have no change in Y with respect to X then it starts to decrease at a phat and then it decreases it even faster rate then it decreases as a faster rate then it starts is still decreasing but it's decreasing at slower and slower rates decreasing it's lower rates and right at that moment it's not you have your slope of your tangent line is 0 then it starts to increase increase so on and so forth and it just keeps happening over and over again so you could also think about this in a more intuitive way