Now we're going to use the graph given here,
to estimate derivatives—slopes of tangent
lines—from the graph.
So it says "Use the given graph to estimate
the value of each derivative.
Hint: think slope of the tangent line.
It may help to draw in the tangent lines."
So the first derivative it asks us to find
for slope is the slope of the tangent line
at negative 3.
So here's negative 3, I draw in a tangent
line here, and, if I draw it well, it looks
like it should be nice and flat.
It's the top of the hill there.
So anytime we have the top of a hill or the
bottom of a hill, like over here at 4, you're
going to have a slope of zero.
Next it asks us to find the slope at negative
one.
So at negative one, if we draw in a tangent
line, here it's harder to be sure, but it
looks like the tangent line looks something
like this.
If I pick some points on this line to look
at, here's negative 1, 1, and here's negative
2, 3, negative 3,5; looks like the change
in Y is about 2, and the change in X is about
1, so this should be 2 over 1, which is 2.
Now if we look at F prime at zero, here is
a very interesting thing.
Because if I put a point here where X is zero,
and I pick any point over here on the right
side to estimate the tangent line it would
look like this.
But if I pick any point over here on the left
side to estimate the tangent line, it's going
to look like this.
As you can see those slopes are going to disagree.
So the limit as H goes to zero from the left
side is not going to equal the limit as H
goes to zero from the right side.
We could write this in derivative notation
saying that F prime from the left side at
zero does not equal F prime from the right
side at zero.
And just like our regular limits, this is
a limit, so if you get two different answers
from two different sides, then the limit does
not exist.
Now let's look at F prime of 1.
At 1, our function is actually a straight
line, and that means that its tangent line
is itself.
So if we just pick any two points on this
line—at 1 it looks like we're at the point
1, one-half, and at 2, it looks like we're
at the point 2, 2 and a half.
That would make our change in Y 2 and our
change in X 1, so we get 2 over 1, which is
2.
Oh and look I've realized we did something
wrong back there in F prime of negative 1.
So we shouldn't have gotten the same answer
at 1 and negative 1 should we?
We have to be careful here, at negative 1,
we should have had a point of negative 2,3,
and a point of negative 1, 1.
The change in Y here is 1 minus 3 is a negative
2, and negative 1 minus [negative] 2 is a
positive 1, so we should have had negative
2 over 1, so negative 2 actually, at negative
1.
And that makes sense, because as you can see
here this tangent line is going downhill here
and should have negative slope.
Alright we have one more tangent line to do,
and that is F prime of 2.
But F prime at 2 is going to have the exact
same problem we had at zero, it has this sharp
corner here, and this sharp corner is going
to tell us that is I take the tangent line,
or if I look at the tangent line on the left
side, and the tangent line on the right side,
their slopes are not going to agree.
So again we get DNE.
What we've learned here is a sharp corner,
called in math a cusp, means that the derivative
there does not exist.
