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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?

• 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