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AP.CALC:

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in the last video we found the slope at a particular point of the curve y is equal to x squared but let's see if we can generalize this and come up with a formula that finds us the slope at any point of the curve y is equal to x squared so let me redraw my function here never hurts to have a nice drawing so that is my y-axis that is my x-axis right there my x-axis let me draw my curve looks something like that you've seen that multiple times this is y is equal to x squared so let's be very general right now remember if we want to find let me just write the definition of our derivative so if we have some point right here let's call that X so we want to be very gentle we want to find the slope at the point X we want to find a function where you give me an X and I'll tell you the slope at that point we're going to call that f prime of x that's going to be the derivative derivative of f of X but all it does is look you when you f of X you give it's a function that you give it an X and it tells you the value of that and we curve we draw the curve here with f of X f of X you give that same X but it's not going to tell you the value of the curve it's not going to say oh this is your f of X it's going to say give you the value of the slope of the curve at that point so f of X if you put it into that function to tell you the oh the slope at that point is equal to you know if you put three there you'll say oh the slope there is equal to six we saw that in the last example so that's what we want to do and we saw in the last I think it was two videos ago that we defined f prime of X we defined F prime of X to be equal to just the well I'll write it this way it's the slope of the secant line between X and some point that's a little bit further away from X so the slope of the secant line is change in Y so it's the F value is the Y value of the point that's a little bit further away from X so f of X plus h minus the Y value at X right because this is right here this is f of X so minus f of X all of that over the change in X so if this is X plus h here the change in X is X plus h minus X or this distance right here is just H the change in X is going to be equal to H so that's your slope of the secant line between any two points like that and we said hey we could find the slope of the tangent line if we just take the limit of this as it approaches as H approaches zero limit as H approaches zero and then we'll be finding the slope of the tangent line now let's apply this idea to a particular function y is or f of X is equal to x squared or Y is equal to x squared so here we could have the point this we could consider this to be the point X x squared so f of X is just equal to x squared and then this would be the point this would be the point let me do it in a more vibrant color this is the point X plus h that's this point right here it's a little bit further down and then X plus h squared X plus h squared and you know in the last video we do this for a particular X we did it for three but now I want a general formula you give me any X and I want it to do what I did in the last video for any particular number I'll have a general function you give me seven I'll tell you what the slope is at seven you give me negative three I'll tell you what the slope is at negative three you give me a hundred thousand I'll tell you what the slope is at a hundred thousand so let's apply it here so we want to find the change in Y over the change in X the change in Y over the change in X so first of all the change in Y the change in Y is this guy's Y value which is X plus h squared X plus h squared that's this guy's Y value right here that's this right here that's X plus h squared I just took X plus h evaluated I squared it and that's that's its point on the curve so it's X plus h squared so that's there right there and then what's this value f of X right here is equal to I know it's getting messy this is equal to x squared if you take your X you evaluate the function at that point you're going to get x squared so it's equal to minus x squared this is your change in Y that's this distance right there change in why I'm just to relate it to our definition of a derivative this blue thing right here is equivalent to this thing right here we just evaluated our function our function is f of X is equal to x squared we just evaluated when X is equal to X plus h so if you have to square it if you have put an A there would be a squared if I put an apple there would be Apple squared I put an X plus h in there it's going to be X plus h squared so this is that thing and then this thing right here is just the function evaluated at the point in question right there so this is our change in Y and let's divide that by our change in X our change in X if this is X plus h and this is just X our change in X is just going to be H so that's where we get that term from so this is just the slope between these two points this is just a slope between those two points but of course we want to find the limit as this point gets closer and closer to this point at this point gets closer and closer to that point so this becomes a tangent line so we're going to take the limit as H approaches zero and this is our f prime of X and this is the exact same definition of this instead of being general and saying for any function we know what the function was it was f of X is equal to x squared so we actually applied it instead of f of X we wrote x squared instead of f of X plus h we wrote X plus h squared so let's see if we can evaluate this limit so this is going to be equal to the limit we write a little neater than that the limit as H approaches 0 to square this out I'll do it in the same color that's x squared plus 2xh plus h squared plus h squared and then we have this minus x squared over here minus x squared I just multiply this guy out over here and then all of that is divided by H now let's see if we can simplify this a little bit well you mean it let's see you have an x squared and you have a minus x squared so those cancel out and then we could divide the numerator and the denominator by H so this simplifies to so we get f prime of X is equal to is equal to if we divide the numerator and denominator by H we get 2x plus h oh sorry I forgot my limit it equals the limit very important limit as H approaches zero of divide the everything by H and you get 2x 2x plus h squared divided by H is H if you remember the last video when we did it with a particular X when we said X is equal to 3 we got 6 plus Delta X here or 6 plus h here so it's very similar so if you take the limit as H approaches 0 here that's just going to disappear so this is just going to be equal to 2x so we just figured out that if f of X this is a big result this is exciting that if f of X is equal to x squared F prime of X is equal to 2x that's what we just figured out and I want to make sure you understand how to interpret this f of X if you give me a value it's going to tell you the value of the function at that point f prime of X is going to tell you the slope at that point let me draw that because this is a this is a key realization and you might you know it's kind of maybe initially unintuitive to think of a function that gives us the slope at any point of another function so it looks like this it looks like let me draw a little neater than that that's still not that neat that's satisfactory let me just draw it in the positive coordinate all that well just draw the hope the curve looks like something like that now this is the curve of f of X this is the curve of f of X is equal to x squared just like that so if you yeah I don't know you give me a point you give me the point 7 you apply you put it in here you square it and it is mapped to the number 49 it is so you get the number 49 right there this is number 749 you're used to dealing with functions right there but what is f prime of 7 F prime of 7 you say 2 times 7 is equal to 14 what is this 14 number here what is this thing well this is the slope of the tangent line at X is equal to 7 so if I were to take that point and draw a tangent line a point a point a point that just grazes our curve if I were to just draw a tangent line that wasn't tangent enough for me so that's my tangent line right there do you get the idea the slope of this guy you do your change in Y over your change in X is going to be equal to 14 the slope of the curve at Y is equal to 7 is a pretty steep curve if you wanted to find the slope let's say that this is y let's say it's X is equal to 2 I said X I said Y at X is equal to 7 the slope is 14 at X is equal to 2 what is the slope well you go you figure out F prime of 2 which is equal to 2 times 2 which is equal to 4 so the slope here so the slope here is 4 the slope is 4 you could say M is equal to 4 M for slope what is f prime of 0 F prime we know that F of 0 is 0 right 0 squared is 0 what is f prime of 0 well 2 times 0 is 0 that's also equal to 0 but what does that mean what's the interpretation it means the slope of the tangent line is 0 so 0 sloped line looks like this looks just like a horizontal line and that looks about right that a horizontal line would be tangent to the curve at y equals 0 let's try another one let's try the point minus - minus 1 so let's say we're right there X is equal to minus 1 so f of minus 1 you just square it because we're dealing with x squared so it's equal to 1 that's that point right there well what is f prime of minus 1 F prime of minus 1 it's 2 times minus 1/2 times minus is minus 2 what does that mean it means that the slope of the tangent line at at X is equal to 1/2 this curve to the function is minus 2 so if I were to draw the tangent line here the tangent line looks like that and look it is a downward sloping line it makes sense the slope here the slope here is equal to minus 2 you

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