2017 AP Calculus AB/BC 4b | AP Calculus AB solved exams | AP Calculus AB | Khan Academy

We're now going to tackle Part B of the potato problem. It says, "Use the second derivative of H with respect to time to determine whether your answer in part A is an underestimate or an overestimate of the internal temperature of the potato at time T equals three."

So, in part A, we found the equation of the tangent line at time equals zero, and we used that to estimate what our internal temperature would be at time equals three. So how is the second derivative going to help us think about whether that was an overestimate or an underestimate?

Well, the second derivative can help us know about concavity. It'll let us know, well, is our slope increasing over this interval, or is our slope decreasing? Then we can use that to estimate whether we overestimated or not.

So first, let's just find the second derivative. We have the first derivative written up here. Let me just rewrite it, and I'll distribute the -1/4 because it'll be a little bit more straightforward then. So if I write the derivative of H with respect to time, it is equal to -4 * our internal temperature, which itself is a function of time, and then - 1/4 * -27 that would be plus 27 over 4.

Let me scroll down a little bit. Now, let me leave that graph up there because I think that might be useful. What is the second derivative going to be with respect to time? So I'll write it right over here. The second derivative of H with respect to time is going to be equal to, well, the derivative of this first term with respect to time is going to be the derivative of this with respect to H times the derivative of H with respect to time.

So this is equal to -1/4; that's the derivative of this term with respect to H. Then we want to multiply that times the derivative of H with respect to time. This comes just straight out of the chain rule, and then the derivative of a constant—how does that change with respect to time? It's not going to change; that is just going to be zero.

So just like that, we were able to find the second derivative of H with respect to time. Now, what does this tell us? Well, we talked about in the previous video that over the interval that we care about—for T greater than zero—it says that our internal temperature is always going to be greater than 27.

So when you look at this expression here, or when you look at this expression here for dH/dt, we talked in the previous video how this is always going to be negative here because H is always going to be greater than 27. So that part's going to be positive. But then we're going to multiply it by -1/4, so our slope dH/dt, our derivative of our temperature with respect to time, is always going to be negative.

So we could write that this—or this—this is going to be negative. Let me write it this way: since H is greater than 27 for T greater than zero, we know that dH/dt is negative.

So we could say that this right over here, since dH/dt is less than zero for T greater than zero, the second derivative of H with respect to T is going to be greater than zero for T greater than zero. So what does that mean? It means that if your second derivative is positive, that means you're concave upward, concave upward, which means slope is increasing.

Slope increasing—or you could say, slope becoming less negative. Now, what does that mean? You can see it intuitively: if the slope is becoming less and less negative, then that means when we approximated what our temperature is at T equals 3, we used a really negative slope when in reality our slope is getting less and less and less negative.

So what we would have done is we would have over-decreased our temperature from T equals 0 to T equals 3. So that would mean that we would have underestimated it.

So let me write that down, and I'm running out of a little bit of space, but let me write it right over here. So that implies—this implies that we underestimated in part A. If I were taking this on the AP exam, I would flesh out my language a little bit here to make it a little bit clearer, but hopefully that makes sense.