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## Connecting a function, its first derivative, and its second derivative

# Calculus-based justification for function increasing

AP.CALC:

FUN‑4 (EU)

, FUN‑4.A (LO)

, FUN‑4.A.10 (EK)

, FUN‑4.A.11 (EK)

## Video transcript

- [Instructor] 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 you know 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, 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? So 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 'cause 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 zero, 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 is it? Are they talking about h? Are they talking about h prime? If they were 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 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.

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