Placeholder Image

字幕列表 影片播放

  • - [Instructor] You are likely already familiar with the idea

  • of a slope of a line.

  • If you're not, I encourage you to review it on Khan Academy,

  • but all it is, it's describing the rate of change

  • of a vertical variable

  • with respect to a horizontal variable,

  • so for example, here I have our classic y axis

  • in the vertical direction and x axis

  • in the horizontal direction,

  • and if I wanted to figure out the slope of this line,

  • I could pick two points,

  • say that point and that point.

  • I could say, "Okay, from this point to this point,

  • what is my change in x?"

  • Well, my change in x would be this distance right over here,

  • change in x,

  • the Greek letter delta, this triangle here.

  • It's just shorthand for "change," so change in x,

  • and I could also calculate the change in y,

  • so this point going up to that point, our change in y,

  • would be this, right over here, our change in y,

  • and then, we would define slope, or we have defined slope

  • as change in y over change in x,

  • so slope is equal to the rate of change

  • of our vertical variable

  • over the rate of change of our horizontal variable,

  • sometimes described as rise over run,

  • and for any line, it's associated with a slope

  • because it has a constant rate of change.

  • If you took any two points on this line,

  • no matter how far apart or no matter how close together,

  • anywhere they sit on the line,

  • if you were to do this calculation,

  • you would get the same slope.

  • That's what makes it a line,

  • but what's fascinating

  • about calculus is we're going to build the tools

  • so that we can think about the rate of change not just

  • of a line, which we've called "slope" in the past,

  • we can think about the rate of change,

  • the instantaneous rate of change of a curve,

  • of something whose rate

  • of change is possibly constantly changing.

  • So for example, here's a curve where the rate of change of y

  • with respect to x is constantly changing,

  • even if we wanted to use our traditional tools.

  • If we said, "Okay, we can calculate the average rate

  • of change," let's say between this point and this point.

  • Well, what would it be?

  • Well, the average rate of change between this point and

  • this point would be the slope of the line

  • that connects them,

  • so it would be the slope of this line of the secant line,

  • but if we picked two different points,

  • we pick this point and this point,

  • the average rate of change

  • between those points all of a sudden looks quite different.

  • It looks like it has a higher slope.

  • So even when we take the slopes between two points

  • on the line, the secant lines,

  • you can see that those slopes are changing,

  • but what if we wanted to ask ourselves

  • an even more interesting question.

  • What is the instantaneous rate of change at a point?

  • So for example, how fast is y changing

  • with respect to x exactly at that point,

  • exactly when x is equal to that value.

  • Let's call it x one.

  • Well, one way you could think about it is

  • what if we could draw a tangent line to this point,

  • a line that just touches the graph right over there,

  • and we can calculate the slope of that line?

  • Well, that should be the rate of change at that point,

  • the instantaneous rate of change.

  • So in this case,

  • the tangent line might look something like that.

  • If we know the slope of this,

  • well then we could say that

  • that's the instantaneous rate of change at that point.

  • Why do I say instantaneous rate of change?

  • Well, think about the video on these sprinters,

  • Usain Bolt example.

  • If we wanted to figure out the speed of Usain Bolt

  • at a given instant, well maybe this describes his position

  • with respect to time if y was position and x is time.

  • Usually, you would see t as time, but let's say x is time,

  • so then, if were talking about right at this time,

  • we're talking about the instantaneous rate,

  • and this idea is the central idea of differential calculus,

  • and it's known as a derivative,

  • the slope of the tangent line, which you could also view

  • as the instantaneous rate of change.

  • I'm putting an exclamation mark

  • because it's so conceptually important here.

  • So how can we denote a derivative?

  • One way is known as Leibniz's notation,

  • and Leibniz is one of the fathers of calculus

  • along with Isaac Newton,

  • and his notation, you would denote the slope

  • of the tangent line

  • as equaling dy over dx.

  • Now why do I like this notation?

  • Because it really comes from this idea of a slope,

  • which is change in y over change in x.

  • As you'll see in future videos,

  • one way to think about the slope

  • of the tangent line is, well,

  • let's calculate the slope of secant lines.

  • Let's say between that point and that point,

  • but then let's get even closer,

  • say that point and that point,

  • and then let's get even closer

  • and that point and that point,

  • and then let's get even closer,

  • and let's see what happens as the change

  • in x approaches zero,

  • and so using these d's instead of deltas,

  • this was Leibniz's way of saying,

  • "Hey, what happens if my changes

  • in, say, x become close to zero?"

  • So this idea,

  • this is known as sometimes differential notation,

  • Leibniz's notation, is instead of just change

  • in y over change in x, super small changes in y

  • for a super small change in x,

  • especially as the change in x approaches zero,

  • and as you will see,

  • that is how we will calculate the derivative.

  • Now, there's other notations.

  • If this curve is described as y is equal to f of x.

  • The slope of the tangent line

  • at that point could be denoted

  • as equaling f prime of x one.

  • So this notation takes a little bit of time getting used to,

  • the Lagrange notation.

  • It's saying f prime is representing the derivative.

  • It's telling us the slope of the tangent line

  • for a given point,

  • so if you input an x into this function into f,

  • you're getting the corresponding y value.

  • If you input an x into f prime,

  • you're getting the slope of the tangent line at that point.

  • Now, another notation that you'll see less likely

  • in a calculus class but you might see in a physics class

  • is the notation y with a dot over it,

  • so you could write this is y with a dot over it,

  • which also denotes the derivative.

  • You might also see y prime.

  • This would be more common in a math class.

  • Now as we march forward in our calculus adventure,

  • we will build the tools to actually calculate these things,

  • and if you're already familiar with limits,

  • they will be very useful, as you could imagine,

  • 'cause we're really going to be taking the limit

  • of our change in y over change in x as our change

  • in x approaches zero,

  • and we're not just going to be able to figure it out

  • for a point.

  • We're going to be able to figure out general equations

  • that described the derivative for any given point,

  • so be very, very excited.

- [Instructor] You are likely already familiar with the idea

字幕與單字

單字即點即查 點擊單字可以查詢單字解釋

B1 中級 美國腔

01-2 導數的概念(Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy)

  • 5 1
    yukang920108 發佈於 2022 年 07 月 12 日
影片單字