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  • - So we have different definitions

  • for d of t on the left and the right

  • and let's say that d is distance and t is time,

  • so this is giving us our distance as a function of time,

  • on the left, it's equal to 3t plus one

  • and you can see the graph of how distance is changing

  • as a function of time here is a line

  • and just as a review from algebra,

  • the rate of change of a line,

  • we refer to as the slope of a line and we can figure it out,

  • we can figure out, well, for any change in time,

  • what is our change in distance?

  • And so in this situation,

  • if we're going from time equal one to time equal two,

  • our change in time, delta t is equal to one

  • and what is our change in distance?

  • We go from distance is equal to four meters,

  • at time equals one,

  • to distance in seven meters at time equal two

  • and so our change in distance here is equal to three

  • and if we wanna put our units,

  • it's three meters for every one second in time

  • and so our slope would be our change in our vertical

  • divided by our change in our horizontal,

  • which would be change in d, delta d over delta t,

  • which is equal to three over one

  • or we could just write that as three meters per second

  • and you might recognize this as a rate,

  • if you're thinking about your change in distance

  • over change in time, this rate right over here

  • is going to be your speed.

  • This is all a review of what you've seen before

  • and what's interesting about a line,

  • or if we're talking about a linear function,

  • is that your rate does not change at any point,

  • the slope of this line between any two points

  • is always going to be three,

  • but what's interesting about this function on the right

  • is that is not true,

  • our rate of change is constantly changing

  • and we're going to study that in a lot more depth,

  • when we get to differential calculus

  • and really this video's a little bit

  • of a foundational primer for that future state,

  • where we learn about differential calculus

  • and the thing to appreciate here

  • is think about the instantaneous rate of change someplace,

  • so let's say right over there,

  • if you ever think about the slope of a line,

  • that just barely touches this graph,

  • it might look something like that,

  • the slope of a tangent line

  • and then right over here,

  • it looks like it's a little bit steeper

  • and then over here, it looks like it's a little bit steeper,

  • so it looks like your rate of change

  • is increasing as t increases.

  • As I mentioned, we will build the tools

  • to later think about instantaneous rate of change,

  • but what we can start to think about

  • is an average rate of change,

  • average rate

  • of change,

  • and the way that we think about our average rate of change

  • is we use the same tools, that we first learned in algebra,

  • we think about slopes of secant lines,

  • what is a secant line?

  • Well, we talk about this in geometry,

  • that a secant is something that intersects a curve

  • in two points, so let's say that there's a line,

  • that intersects at t equals zero and t equals one

  • and so let me draw that line, I'll draw it in orange,

  • so this right over here is a secant line

  • and you could do the slope of the secant line

  • as the average rate of change

  • from t equals zero to t equals one,

  • well, what is that average rate of change going to be?

  • Well, the slope of our secant line is going to be

  • our change in distance divided by our change in time,

  • which is going to be equal to,

  • well, our change in time is one second,

  • one, I'll put the units here, one second

  • and what is our change in distance?

  • At t equals zero or d of zero is one

  • and d of one is two,

  • so our distance has increased by one meter,

  • so we've gone one meter in one second

  • or we could say that our average rate of change

  • over that first second from t equals zero,

  • t equals one is one meter per second,

  • but let's think about what it is,

  • if we're going from t equals two

  • to t equals three.

  • Well, once again, we can look at this secant line

  • and we can figure out its slope,

  • so the slope here, which you could also use

  • the average rate of change

  • from t equals two to t equals three,

  • as I already mentioned,

  • the rate of change seems to be constantly changing,

  • but we can think about the average rate of change

  • and so that's going to be our change in distance

  • over our change in time,

  • which is going to be equal to when t is equal to two,

  • our distance is equal to five,

  • so one, two, three, four, five,

  • so that's five right over there

  • and when t is equal to three, our distance is equal to 10,

  • six, seven, eight, nine, 10, so that is 10 right over there,

  • so our change in time, that's pretty straightforward,

  • we've just gone forward one second, so that's one second

  • and then our change in distance right over here,

  • we go from five meters to 10 meters is five meters,

  • so this is equal to five meters per second

  • and so this makes it very clear,

  • that our average rate of change has changed

  • from t equals zero, t equals one

  • to t equals two to t equals three,

  • our average rate of change is higher

  • on this second interval, than on this first one

  • and as you can imagine,

  • something very interesting to think about

  • is what if you were to take the slope

  • of the secant line of closer and closer points?

  • Well, then you would get closer and closer

  • to approximating that slope of the tangent line

  • and that's actually what we will do when we get to calculus.

- So we have different definitions

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平均變化率介紹|函數|代數I|可汗學院 (Introduction to average rate of change | Functions | Algebra I | Khan Academy)

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    piano.man 發佈於 2021 年 01 月 14 日
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