斜率與變化速率 (英)

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斜率與變化速率 (英) : 斜率與變化速率

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In this video I'm going to do a bunch of example slope
problems. Just as a bit of review, slope is just a way of
measuring the inclination of a line.
And the definition-- we're going to hopefully get a good
working knowledge of it in this video-- the definition of
it is a change in y divided by change in x.
This may or may not make some sense to you right now, but as
we do more and more examples, I think it'll make a good
amount of sense.
Let's do this first line right here.
Line a.
Let's figure out its slope.
They've actually drawn two points here that we can use as
the reference points.
So first of all, let's look at the
coordinates of those points.
So you have this point right here.
What's its coordinates?
Its x-coordinate is 3.
Its y-coordinate is 6.
And then down here, this point's x-coordinate is
negative 1 and its y-coordinate is negative 6.
So there's a couple of ways we can think about slope.
One is, we could look at it straight up using the formula.
We could say change in y-- so slope is change in y over
change in x.
We can figure it out numerically.
I'll in a second draw it graphically.
So what's our change in y?
Our change in y is literally how much did our y values
change going from this point to that point?
So how much did our y values change?
Our y went from here, y is at negative 6 and it went all the
way up to positive 6.
So what's this distance right here?
It's going to be your end point y value.
It's going to be 6 minus your starting point y value.
Minus negative 6 or 6 plus 6, which is equal to 12.
You could just count this.
You say one, two, three, four, five, six, seven, eight, nine,
ten, eleven, twelve.
So when we changed our y value by 12, we had to change our x
value by-- what was our change it x over the
same change in y?
Well we went from x is equal to negative 1 to
x is equal to 3.
Right? x went from negative 1 to 3.
So we do the end point, which is 3 minus the starting point,
which is negative 1, which is equal to 4.
So our change in y over change in x is equal to 12/4 or if we
want to write this in simplest form, this is the
same thing as 3.
Now the interpretation of this means that for every 1 we move
over-- we could view this, let me write it this way.
Change in y over change in x is equal to-- we could say
it's 3 or we could say it's 3/1.
Which tells us that for every 1 we move in the positive
x-direction, we're going to move up 3 because this is a
positive 3 in the y-direction.
You can see that.
When we moved 1 in the x, we moved up 3 in the y.
When we moved 1 in the x, we moved up 3 in the y.
If you move 2 in the x-direction, you're going to
move 6 in the y.
6/2 is the same thing as 3.
So this 3 tells us how quickly do we go up as we increase x.
Let's do the same thing for the second line on this graph.
Graph b.
Same idea.
I'm going to use the points that they gave us.
But really you could use any points on that line.
So let's see, we have one point here, which is
the point 0, 1.
You have 0, 1.
And then the starting point-- we could call this the finish
point-- the starting point right here, we could view it
as x is negative 6 and y is negative 2.
So same idea.
What is the change in y given some change in x?
So let's do the change in x first. So what is
our change in x?
So in this situation, what is our change in x? delta x.
We could even count it.
It's one, two, three, four, five, six.
It's going to be 6.
But if you didn't have a graph to count from, you could
literally take your finishing x-position, so it's 0, and
subtract from that your starting x-position.
0 minus negative 6.
So when your change in x is equal to-- so this will be 6--
what is our change in y?
Remember we're taking this as our finishing position.
This is our starting position.
So we took 0 minus negative 6.
So then on the y, we have to do 1 minus negative 2.
1 minus negative 2
What's 1 minus negative 2?
That's the same thing as 1 plus 2.
That is equal to 3.
So it is 3/6 or 1/2.
So notice, when we moved in the x-direction by 6, we moved
in the y-direction by positive 3.
So our change in y was 3 when our change in x was 6.
Now, one of the things that confuses a lot of people is
how do I know what order to-- how did I know to do the 0
first and the negative 6 second and then the 1 first
and then the negative 2 second.
And the answer is you could've done it in either order as
long as you keep them straight.
So you could have also have done change in y
over change in x.
We could have said, it's equal to negative 2 minus 1.
So we're using this coordinate first. Negative 2 minus 1 for
the y over negative 6 minus 0.
Notice this is a negative of that.
That is the negative of that.
But since we have a negative over negative, they're going
to cancel out.
So this is going to be equal to negative 3 over negative 6.
The negatives cancel out.
This is also equal to 1/2.
So the important thing is if you use this y-coordinate
first, then you have to use this
x-coordinate first as well.
If you use this y-coordinate first, as we did here, then
you have to use this x-coordinate
first, as you did there.
You just have to make sure that your change in x and
change in y are-- you're using the same final
and starting points.
Just to interpret this, this is saying that for every minus
6 we go in x.
So if we go minus 6 in x, so that's going backwards, we're
going to go minus 3 in y.
But they're essentially saying the same thing.
The slope of this line is 1/2.
Which tells us for every 2 we travel in x, we go up 1 in y.
Or if we go back 2 in x, we go down 1 in y.
That's what 1/2 slope tells us.
Notice, the line with the 1/2 slope, it is less steep than
the line with a slope of 3.
Let's do a couple more of these.
Let's do line c right here.
I'll do it in pink.
Let's say that the starting point-- I'm just picking this
arbitrarily.
Well, I'm using these points that they've drawn here.
The starting point is at the coordinate negative 1, 6 and
that my finishing point is at the point 5, negative 6.
Our slope is going to be-- let me write this-- slope is going
to be equal to change in x-- sorry, change in y.
I'll never forget that.
Change in y over change in x.
Sometimes it's said rise over run.
Run is how much you're moving in the horizontal direction.
Rise is how much you're moving in the vertical direction.
Then we could say our change in y is our finishing y-point
minus our starting y-point.
This is our finishing y-point.
That's our starting y-point, over our finishing x-point
minus our starting x-point.
If that confuses you, all I'm saying is, it's going to be
equal to our finishing y-point is negative 6 minus our
starting y-point, which is 6, over our finishing x-point,
which is 5, minus our starting x-point, which is negative 1.
So this is equal to negative 6 minus 6 is negative 12.
5 minus negative 1.
That is 6.
So negative 12/6.
That's the same thing as negative 2.
Notice we have a negative slope here.
That's because every time we increase x by 1, we go down in
the y-direction.
So this is a downward sloping line.
It's going from the top left to the bottom right.
As x increases, the y decreases.
And that's why we got a negative slope.
This line over here should have a positive slope.
Let's verify it.
So I'll use the same points that they
use right over there.
So this is line d.
Slope is equal to rise over run.
How much do we rise when we go from that point to that point?
Let's see.
We could do it this way.
We are rising-- I could just count it out.
We are rising one, two, three, four, five, six.
We are rising 6.
How much are we running?
We are running-- I'll do it in a different color.
We're running one, two, three, four, five, six.
We're running 6.
So our slope is 6/6, which is 1.
Which tells us that every time we move 1 in the x-direction--
positive 1 in the x-direction-- we go positive 1
in the y-direction.
For every x, if we go negative 2 in the x-direction, we're
going to go negative 2 in the y-direction.
So whatever we do in x, we're going to do the same thing in
y in this slope.
Notice, that was pretty easy.
If we wanted to do it mathematically, we could
figure out this coordinate right there.
That we could view as our starting position.
Our starting position is negative 2, negative 4.
Our finishing position is 4, 2.
4,2
So our slope, change in y over change in x.
I'll take this point 2 minus negative 4 over 4 minus
negative 2.
2 minus negative 4 is 6.
Remember that was just this distance right there.
Then 4 minus negative 2, that's also 6.
That's that distance right there.
We get a slope of 1.
Let's do another one.
Let's do another couple.
These are interesting.
Let's do the line e right here.
Change in y over change in x.
So our change in y, when we go from this point to this
point-- I'll just count it out.
It's one, two, three, four, five, six, seven, eight.
It's 8.
Or you could even take this y-coordinate 2 minus negative
6 will give you that distance, 8.
What's the change in y?
Well the y-value here is-- oh sorry what's the change in x?
The x-value here is 4.
The x-value there is 4.
X does not change.
So it's 8/0.
Well, we don't know.
8/0 is undefined.
So in this situation the slope is undefined.
When you have a vertical line, you say
your slope is undefined.
Because you're dividing by 0.
But that tells you that you're dealing probably with a
vertical line.
Now finally let's just do this one.
This seems like a pretty straight up vanilla slope
problem right there.
You have that point right there, which is
the point 3, 1.
So this is line f.
You have the point 3, 1.
Then over here you have the point negative 6, negative 2.
So our slope would be equal to change in y.
I'll take this as our ending point, just so you can go in
different directions.
So our change in y-- now we're going to go
down in that direction.
So it's negative 2 minus 1.
That's what this distance is right here.
Negative 2 minus 1, which is equal to negative 3.
Notice we went down 3.
And then what is going to be our change in x?
Well, we're going to go back that amount.
What is that amount?
Well, that is going to be negative 6, that's our end
point, minus 3.
That gives us that distance, which is negative 9.
For every time we go back 9, we're going to go down 3.
If we go back 9, we're going to go down 3.
Which is the same thing as if we go forward 9, we're going
to go up 3.
All equivalent.
And we see these cancel out and you get a slope of 1/3.
Positive 1/3.
It's an upward sloping line.
Every time we run 3, we rise 1.
Every time we run 3, we rise 1.
Anyway, hopefully that was a good review of slope for you.