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# Connecting f, f', and f'' graphically

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

## Video transcript

we have the graphs of three functions here and what we know is that one of them is the function f another is the first derivative of F and then the third is the second derivative of F and our goal is to figure out which function is which which one is f which is the first derivative and which is a second like always pause this video and see if you can work through it on your own before we do it together all right now let's do this together the way I'm going to tackle it is I'm going to try to sketch what we can about the derivatives of each of these graphs or each of the functions represented by these graphs so in this first graph here in this orange color we can see that the slope is quite positive here but then it becomes less and less and less positive up until this point where the slope is going to be 0 and then it becomes more and more and more and more negative so the derivative of this curve right over here or the function represented by this curve it's gonna start off reasonably positive right over there at this point the derivative is gonna cross 0 because our derivative is 0 they our slope of the tangent line and then it's going to get more and more negative or at least over the interval that we see so it might look I don't know something like this I don't know if it's a line or not it might be some type of a curve but it go it might go it would definitely have a trend something like that now we can immediately tell that this blue graph is not the derivative of this orange graph its trend is opposite over that interval it's going from being negative to positive as opposed to be going from positive to negative so we can rule out the blue graph as being the derivative of the orange graph but what about this magenta graph it does have right it does look like it has the right trend in fact it intersects the x-axis at the right place right over there and at least over this interval it seems it's positive from here to here so it's positive this graph is positive when the slope of the tangent line here is positive and this graph is negative when the slope of the tangent line here is negative now one thing that might be causing some unease to immediately say that this last graph is the derivative of the first one is we're not used to situations where has more extreme points more minima and maxima than the original function but in this case it could just be because we don't see the entire original function so for example if this last graph is indeed the derivative of this first graph then what we see is our derivative is our graph our derivative is negative right over here but then right around here it starts becoming less negative so that point corresponds to roughly right over there then over here our slope will become become less and less and less negative and then at this point our slope would become zero which would be right around there so for example our graph might look something like this we just didn't see it it fell off of the part of the graph that we actually showed so I would actually say that this is a good candidate for being the third function is a good candidate for being the derivative of the first function so maybe we could say that this is f and that this is f Prime now let's look at the second graph what would its derivative look like so over here our slope is quite negative and it becomes less and less and less negative until we go right over here where our slope is zero so our derivative would intersect the x-axis right over there it would start out negative and it would become less and less and less negative and at this point it crosses the x-axis and then it becomes more and more positive so we see here our derivative becomes more and more positive but then right around here it seems like it's getting less positive again so it might look something might look something like this where over here it's becoming less positive again less positive less positive less positive right over here our derivative would be zero so we our derivative would intersect the x-axis there and then it just looks like it is the slope is getting more and more and more negative so our dream is going to get more and more and more negative well what I very roughly just sketched out looks an awful lot like the brown graph right over here so this brown graph does indeed look like the derivative of this blue graph so what I would say is that this is actually f and then this would be f prime and then if this is f prime the derivative of that is to be F prime prime so that looks good I would actually go with this and if you wanted just for safe measure you could try to sketch out what the derivative of this graph would be and actually let's just do that so over here the derivative of this so right now we have a positive slope of our tangent line is getting less and less positive it hits zero right over there so the derivative might look something like this over that interval now the slope of the tangent line is getting more and more and more and more negative right until about that point so it's getting more and more and more negative until about that point and now it looks like it's getting less and less and less and less negative all the way until the derivative goes back to being zero and then it looks like it's getting more and more and more and more positive so the derivative of this magenta curve looks like an upward-opening you and we don't see that over here so we could feel good that it's derivative actually isn't depicted so I feel good calling the middle graph F they're calling the left graph F prime and calling the right graph the second derivative