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# 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

we are told the differentiable function H and its derivative H prime or graph and you can see it here H is in blue and then it's derivative H prime is in this orange color for 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 teachers 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 and so maybe use the derivative in some way now we 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 0 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 teachers comments to the justifications so one student wrote the derivative of H is increasing when X is greater than 0 so it is indeed the case that the derivative is increasing when X is greater than 0 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's that is a justification for why H is increasing but that's not calculus-based and 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 is are they talking about H 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 corrected 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