Name: 01-3 割線和平均變化率（Secant lines & average rate of change | Derivatives introduction | AP Calculus AB | Khan Academy）
Uploaded: 2022-07-12T11:56:08.000Z
Duration: 5 min 17 s
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過去簡單式

we have the graph of y is equal to x squared,

or at least part of the graph of y is equal to x squared.

And the first thing I'd like to tackle is

over the interval from x equaling 1 to x equaling 3.

We want to know the average rate of change

over the interval from x going from 1 to 3.

where x could be 1, and x could be equal to 3.

Well we could do this even without looking at the graph.

where, if this is x, and this is y is equal to x squared,

when x is equal to 1, y is equal to 1 squared,

y is equal to 3 squared, which is equal to 9.

And to figure out the average rate of change

you say, "Okay, well what's my change in x?"

Well, what's our change in y over the same interval?

y increases by 8, so it's going to be a positive 8.

which is equal to 8 over 2, which is equal to 4.

So that would be our average rate of change.

every time x increases by 1, y is increasing by 4.

and we calculated change in y over change of x

Now this might be looking fairly familiar to you,

as the slope of a line connecting two points.

If you were to draw a secant line between these two points,

And so the average rate of change between two points,

that is the same thing as the slope of the secant line.

in comparison to the curve over that interval,

it hopefully gives you a visual intuition

for what even average rate of change means.

Because in the beginning part of the interval,

is increasing at a faster rate than the secant line,

And so that's why the slope of the secant line

Is it the exact rate of change at every point?

The curve's rate of change is constantly changing.

and then it's actually increasing at a higher rate

the change in y over the change in x is exactly the same.

Now one question you might be wondering is

why are you learning this is in a calculus class?

Couldn't you have learned this in an algebra class?

and is really one of the foundational ideas of calculus is

We found the average rate of change between 1 and 3,

or the slope of the secant line from (1, 1) to (3, 9).

But what instead if you found the slope of the secant line

But what if you wanted to get even closer?

Let's say you wanted to find the slope of the secant line

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01-3 割線和平均變化率（Secant lines & average rate of change | Derivatives introduction | AP Calculus AB | Khan Academy）

constantly

essentially

eventually

absolutely