In this section, we want to explore the connection between the graph of a function,

the graph of it’s first derivative, and the graph of it’s second derivative.

How do all three of those graphs relate?

And our goal at the end of this,

is to be able to sort through a bunch of different looking graphs,

and be able to identify which is the f, which is f’, and which is f’’.

Now, let’s begin by taking the familiar function y = x^3, and graphing it.

And then, let’s ask the simple question:

What can we tell about the first and second derivatives, from this graph?

Now concerning the first derivative,

whatever the slope of the function is at any point,

that’s the value of the first derivative there.

Now just before we move on,

in order to make sure you’re solid with that aspect of things,

why don’t you try the following activity which tries to match first derivatives with their function.

To do it, click and drag each first derivative to its correct original function. Have fun!

Great!! Now, the next question we want to answer is:

what does the second derivative look like?

Well, consider the following function:

And then let’s use the same method we just used to get the graph of the first derivative.

And now, how can we use this to get the second derivative?

Well, I suppose one way we could do it, and to be honest,

probably the way I’d recommend,

is to repeat the process we just did,

but this time on the first derivative instead of the original function.

Because after all, the second derivative, is just the derivative of the first derivative.

So let’s do that, let’s get the slope at every point of our first derivative,

and then plot that to get our second derivative.

Now in a moment,

you’re going to be given a bunch of graphs and asked to match both the first and second derivatives with their correct original function,

and doing the above method is a solid way to go.

But let me just give you one extra tip to remember when helping to identify the second derivative.

Do you remember what the second derivative always tells us?

It tells us the concavity. So for instance, on the original function,

notice that on this part of the curve, it’s concave down the entire time,

that means the 2nd derivative must be negative,

and so we can expect if we're graphing the 2nd derivative and since it's negative for it to be BELOW the x-axis for this entire interval,

just as it is, and for the other part of the original curve, it’s concave UP,

the entire time, and so the 2nd derivative must be POSITIVE.

And so we’d expect the graph of the 2nd derivative because it's POSITIVE to be ABOVE the x-axis on that interval, just as it is.

And so, it’s time to put it all into practice.

Try the following activity and see if you can correctly match each first derivative and second derivative with the original function.

Have fun!

Great job!!! And that’s it for this lesson!