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## Derivative as a function

Current time:0:00Total duration:3:56

# Matching functions & their derivatives graphically (old)

## Video transcript

The function f of x
is shown in green. The sliding purple window
may contain a section of an antiderivative of
the function, F of x. So, essentially it's
saying, this green function, or part of this green
function, is potentially the derivative of
this purple function. And what we need
to do is-- it says, where does the function
in the sliding window correspond to the
antiderivative of our function? The antiderivative of f of x,
usually, write as big F of x. This is just saying
that, lowercase f of x is just the derivative
of big F of x. So, at what point
could the derivative of the purple
function-- and I'm going to move the purple
function around-- where can the derivative of
that be the green stuff. So let's just focus on
the purple stuff first. So the derivative--
we can just view it as the slope of the tangent
line-- between this point and this point, we see that we
have a constant negative slope, and then we have a
constant positive slope. So let's see, where here do we
have a constant negative slope? Well, now here the slope is
0, and it gets more negative. Here we have a constant
positive slope, not a constant negative slope. Here we have a constant
negative slope, so maybe it matches
up over there. So here we have a
constant negative slope, but then on the purple function,
we have a positive slope, but where the potential
derivative is here, we just have a slope of 0. So, this doesn't
match up either. So it looks like in this case,
there's actually no solution. Let's see if this works out. Yes, correct. Next question. Let's do another one. A function f of x
is shown purple. The sliding green
window may contain a section of its derivative. So now we're trying
to say, at what point of this purple
function might the derivative look like
this green function? So in this green
function, if this is the function's
derivative, here the slope is very negative. It goes to 0, and then
the slope gets positive. So let's think about it. So over here, the slope is
just a constant negative, so that won't work. If we shift it over
here, our slope is very steep in the
negative direction and then it gets
less and less steep in the negative direction,
and it goes all the way, and then over here
the slope is 0. And over here, if this
is the derivative, it seems to match
up, the slope is 0. And then it gets
more and more steep in the positive direction. So this matches up. It looks like over
this interval, the green the function
is indeed the derivative of this purple function. So let's see. Let's check our answer. Correct. Next question. Let's do another one. This is exciting. A function f of x
is shown in green. The sliding purple window
may contain a section of an antiderivative of
the function, F of x. So, now we say, let's match
up this little purple section to its derivative. So the green is the
derivative, the purple function is the thing we're
taking the derivative of. So if we just look
at the purple, we see that we have a
constant negative slope in the first part of
it, then our slope-- so let me just look for where I can
find a constant negative slope. So here, this is a
constant positive slope. This is not a constant slope. This is a constant positive. Here's a constant
negative slope. Let's see if this works. So over this interval, between
here and here, my slope is a constant negative,
and indeed, it looks like a constant negative. And you see it's a
constant negative 1. And over here, you
see the derivative is right at negative
1, and it's constant, so that part looks good. And then when I look
at the purple function, my slope is 0
starting off, then it gets more and more steep
in the negative direction. And so my slope is 0, and it
gets more and more negative, so this is indeed
seems to match up. So, let's check our answer. Yes, got it right. I could keep doing this. This is so much fun.