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.

now do it with integrals ðŸ˜›

This is exactly what I asked for it will help me study for my ab midterm

Can you do even harder visualizing derivative and antiderivative problems

Really great to start off visualizing this way, then moving into the actual math of it second!

Just in time for finals! ðŸ˜€

It really is a fun exercise. And I could see how to easily do variations of it to make versions of it quite a bit more challenging. For instance, you could introduce exponential functions, roots, logarithms and higher powers.

i thought this was going to be about stock options… I have been studying too much finance -__-"