Calculus based justification for function increasing | AP Calculus AB | Khan Academy

We are told the differentiable function h and its derivative h prime are graphed, and you can see it here. h is in blue, and then its derivative h prime is in this orange color. Four students were asked to give an appropriate calculus-based justification for the fact that h is increasing when x is greater than zero. Can you match the teacher's comments to the justifications?

So, before I even look at what the students wrote, you might say, "Hey, look, I can just look at this and see that h is increasing when x is greater than zero." But just by looking at the graph of h, that by itself is not a calculus-based justification. We're not using calculus; we're just using our knowledge of what it means for a graph to be increasing.

In order for it to be a calculus-based justification, we should use calculus in some way, so maybe use the derivative in some way. Now, you might recognize that a function is increasing if its derivative is positive. So before I even look at what the students said, what I would say, my calculus-based justification—and I wouldn't even have to see the graph of h—I would just have to see the graph of h prime, is to say, "Look, h prime is greater than zero; h prime is positive when x is greater than zero. If the derivative is positive, then that means that the slope of the tangent line is positive, and that means that the graph of the original function is going to be increasing."

Now let's see whether one of the students said that or what some of the other students wrote. So, can you match the teacher's comments to the justifications? One student wrote, "The derivative of h is increasing when x is greater than zero." So it is indeed the case that the derivative is increasing when x is greater than zero, but that's not the justification for why h is increasing.

For example, the derivative could be increasing while still being negative, in which case h would be decreasing. The appropriate justification is that h prime is positive—not that it's necessarily increasing, because you could be increasing and still not be positive. So let's see. I would say that this doesn't justify why h is increasing when x is greater than 0.

As the x values increase, the function values also increase. Well, that is a justification for why h is increasing, but that's not calculus-based. In no way are you using a derivative, so this isn't a calculus-based justification. It's above the x-axis! So this one... what are they—what is it? Are they talking about h, or are they talking about h prime? If they're saying that h prime is above the x-axis when x is greater than zero, then that would be a good answer. But this is just... you know, what is above the x-axis and over what interval?

So I would actually... let's scroll down a little bit. This looks like a good thing for the teacher to write: "Please use more precise language; this cannot be accepted as a correct justification."

And then finally, this last student wrote, "The derivative of h is positive when x is greater than zero," and it is indeed the case: if your derivative is positive, that means that your original function is going to be increasing over that interval. So kudos, you are correct!