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# Connecting f, f', and f'' graphically (another example)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.10 (EK)
,
FUN‑4.A.11 (EK)

## Video transcript

we have the graph of three functions here and we're told that one of them is the function f one is its first derivative and then one of them is the second derivative we just don't know which one is which and so like always pause this video and see if you can figure it out alright now the way I'm going to tackle it is I'm gonna look at each of these graphs and try to think what would their derivatives look like so for this first one we can see our derivative right over here is our slope of our tangent line it would be a little bit negative and then it gets more and more and more negative and as we approach this vertical asymptote right over here it looks like it's approaching negative infinity so the derivative would actually over here it would be a little bit less than zero but then it would get more and more and more negative and then it would approach negative infinity so it would have a similar shape general shape to the graph itself at least to the left of this vertical asymptote now what about to the right of the vertical asymptote right to the right of the vertical asymptote it looks like the slope of the tangent line is very negative it's very negative but then it becomes less and less and less less and less and less negative and it looks like it is approaching it is approaching zero so on this side the derivative starts out super negative and then it looks like it is the derivative is going to asymptote towards zero something like that so based on what we just it actually looks like so based on what I just sketched it just looks like this right graph is a good candidate for the derivative of this left graph you might say what's wrong with this blue graph well this blue graph out here notice it's positive so if this were the derivative of the left graph then that means that the left graph would need a positive slope out here but it doesn't have a positive slope it's a very it's a slightly negative slope becoming super negative and so right here we're slightly negative and then we become very negative and so maybe this is let's call let's call this F and maybe this is f prime this is f prime right over here and now let's look at this middle graph what would its derivative do so over here our slope is slightly negative and then it becomes more and more and more and more and more negative and so the derivative of this might look like it has to be it has to be slightly negative but then it gets more and more and more and more negative as we approach that vertical asymptote and on the right side of the vertical asymptote our derivative is very positive here and then it gets less and less and less and less and less and less positive and so we start our derivative would be very positive and then it would get less and less and less and less positive it looks like it might the slope here might be asymptoting towards zero so our graph might look something like that well the left graph right here looks a lot like what I just sketched out as a candidate derivative for this blue graph for this middle graph and so I would say that this is F then this is the derivative of that which would make it f Prime and then we already established that this right graph is the derivative of the left one so if it's the derivative of F prime it's not f prime itself it's the second derivative so I feel pretty good about that and just for good measure we could think about what the derivative of this graph would look like here the slope is slightly negative but then it gets more and more and more and more and more negative so the derivative would have a similar shape here and then here our derivative would be very positive and it gets less and less and less and less and less positive and so we start very positive and then it gets less and less and less and less and less positive and so as a general shape it actually does look a lot like this first graph but the reason why I'm not going to say that this first graph is the derivative of the right hand graph is because this right hand graph was the only good candidate that we had for the derivative of the left hand graph and so I feel pretty good with what we selected that this middle one is f the left one is the first derivative and the right one is the second derivative