Slope

Only about ten percent of the Amazon's water enters downstream of Óbidos, very little of which is from the northern slope of the valley.

That's the slope of the slope, or the rate of change of the slope itself.

The sloping roof.

The slope was increasing.

Slope decreases.

They have come to the top of a cliff.

Valley Center!

So our slope has decreased.

Slope is decreasing.

The slope was very positive.

The rate of change of the slope is negative.

Throw me off the cliff!

New Valley

Cavalry in the valley.

...and the valley beyond. Oh.

One, they said, Life is on an incline.

We made the last one. I know.

Yes, for over there on the steep side.

It's in the West Valley.

We went down into the Rift Valley.

You go back to the rim of the gorge.

I watched it clear through the valley.

I'll show you the fastest way over.

A mist settled over the valley.

So what did we say about concave upwards? concave upwards is when the slope is increasing, which means that the rate of change of the slope is positive.

If the slope is constantly increasing, then the rate of change of slope is positive, then the second derivative is positive.

If the slope is decreasing, then the rate of change of slope is negative, which tells us that the second derivative is negative.

If only we had some sort of ramp... ding

So as we increased x, the slope increased.

And as we increase now, our slope becomes negative.

Our slope was decreasing up until this point.

Now, in concave downwards, the slope is decreasing.

A time when you can't tell between the road and a cliff.

It would have been so easy. Remember the precipice?

There are dizzy daffodils on the hillside

Then we come to the rapids?

Hendricks and his men are headed for the rapids.

Fiske, Vicente, watch that end of the ridge.

You spawn of the devil's own strumpet!

Then went I up in the night by the brook, and viewed the wall and I turned back, and entered by the valley gate, and so returned.

Then went I up in the night by the brook, and viewed the wall, and turned back, and entered by the gate of the valley, and so returned.

I liked the room overlooking the valley.

Little more than anyone else in this wretched valley.

Let's do Lovely Lily of the Valley.