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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.