U2S1V04 Approximation and Tangent Lines.txt

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Here's our boat.
We had two pieces of information about it.
We knew that its position was 150 meters at t
equals 20 seconds, and its velocity
was 0.4 meters per second at that same time.
From that, we wanted to approximate the position at t
equals 30, 10 seconds later.
One way you might have thought about that
would have been to make a simplifying assumption,
that the boat kept the same velocity over the 10 seconds.
In which case, you take the 0.4 meters per second
and multiply it by the 10 seconds
to get an estimate of 4 meters as the amount the boat traveled
or the difference in position.
So our best guess would then be 154 meters
as the boat's position at time t equals 30.

Let's see what this has to do with tangent lines.
Our position at time t is given by the function x of t.
And we know two things about x.
First, we know that x of 20 is 150.
So that means the 0.20, 150 is on the graph of x.
And the second thing we know is that x
prime of 20-- the velocity-- that's 0.4 meters per second.
And that's the slope of the tangent line at this point.
We don't know anything else about the graph.
The graph could look like this or it could look like this.
The only thing we know from this information
is that it's tangent to this line at this point.
But that's actually quite useful,
because we know that a tangent line is
going to be really close to the graph near this point.
So we can approximate x of 30, whether that's
here or here, by the height of the tangent line at t
equals 30.
How high is that?
Well starting from this point, which is at height 150,
we've gone over by 10 and the slope is 0.4.
So we've gone up by 0.4 times 10, which
is 4, and so we end up at 154.
This is the exact same calculation
that we did previously with the velocity.
We decided to treat the boat as if it
had constant velocity, which is the same thing as treating
the graph as if it's a straight line with constant slope.
Now the great thing about thinking
of this in terms of a tangent line and not just velocity,
is the tangent lines work in a lot more contexts.
You don't have to have something physical moving.
So we've got a question for you where you can try and think
about just how this applies in a more abstract setting.
So why don't you take a moment and think about that.