The graphical relationship between a function & its derivative (part 1) (video) | Khan Academy
The Ratio and Root Tests · 8. Power Series . We will begin to use different notations for the derivative of a function. .. In order for the notion of the tangent line at a point to make sense, the curve must be "smooth'' at that point. . The absolute value function has no tangent line at 0 because there are (at least) two obvious. The Concept of Derivative Â· A Discontinuous Function - the Step Function Â· This insures that the graph of the function conforms exactly with the above definition. . for a straight line because the ratio of the rise to the run is always constant. The smaller the absolute value of h, the closer we get to our derivative. This is a HUGE topic to cover properly. I think it would be a good idea just to consider some simple polynomial graphs. Notice what happens to.
If the independent variable happens to be "time", we often think of this ratio as a rate of change an example is velocity If we zoom in on the graph of the function at some point so that the function looks almost like a straight line, the derivative at that point is the slope of the line. This is the same as saying that the derivative is the slope of the tangent line to the graph of the function at the given point. The slope of a secant line line connecting two points on a graph approaches the derivative when the interval between the points shrinks down to zero.
So what is the derivative, after all?
The derivative is also, itself, a function: For example, the velocity of a car may change from moment to moment as the driver speeds up or slows down. The last remark is quite important and interesting: We could then talk about its derivative!
Ofcourse, we do this very often without realizing it! Whenever we talk about acceleration we are talking about the derivative of a derivative, i. Second derivatives and third derivatives, and so on are also functions!
Each one tells us about the rate of change of the previous function in this pyramid scheme. We have used a lot of words to try to describe what the derivative is. Mathematicians try to avoid lots of words, aiming at precision and succinctness.
Let's take a look at what they might do instead. A mathematician's code Mathematicians have developed a kind of "secret code" that says all of the things we have enumerated above with a few strokes of the pen.
It actually took centuries to develop this code to the point where it became part of the mathematical society's accepted language, but now it is used universally. So we can see when x is equal to negative 4, the slope of the tangent line is essentially vertical.
The graphical relationship between a function & its derivative (part 1)
So you could say it's not really defined there. But as we go slightly to the right of x equals negative 4, we just have a very, very, very positive slope.
So you could kind of view it as our slope is going from infinity to very, very positive to a little bit less positive to a little bit less positive, to a little bit less positive, to a little bit less positive.
So which of these graphs here have that property? Remember, this is trying to graph the slope. So which of these functions down here, which of these graphs, have a value that is essentially kind of approaching infinity when x is equal to negative 4, and then it gets less and less and less positive as x goes to 0?
So this one, it looks like it's coming from negative infinity, and it's getting less and less and less negative. So that doesn't seem to meet our constraints.
What is the relationship between the graph of a function and the graph of its derivative? - Quora
This one looks like it is coming from positive infinity, and it's getting less and less and less positive, so that seems to be OK. And so when you look at the derivative, the slope is still a positive value. But as we get larger and larger x's up to this point, the slope is getting less and less positive, all the way to 0. And then the slope is getting more and more negative.
And at this point, it seems like the slope is just as negative as it was positive there.
So at this point right over here, the slope is just as negative as it was positive right over there. So it seems like this would be a reasonable view of the slope of the tangent line over this interval. Now let's think about as we get to this point. Here the slope seems constant. Our slope is a constant positive value. So once again, our slope here is a constant positive line. Let me be careful here because at this point, our slope won't really be defined, because our slope, you could draw multiple tangent lines at this little pointy point.
So let me just draw a circle right over there. But then as we get right over here, the slope seems to be positive. So let's draw that. The slope seems to be positive, although it's not as positive as it was there. So the slope looks like it is-- I'm just trying to eyeball it-- so the slope is a constant positive this entire time.
We have a line with a constant positive slope. So it might look something like this.
Matching functions & their derivatives graphically
And let me make it clear what interval I am talking about. I want these things to match up. So let me do my best. So this matches up to that.