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# Tangents of polynomials

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.3 (EK)

## Video transcript

what you see here in blue this is the graph of y is equal to f of X where f of X is equal to X to the third minus 6x squared plus X minus 5 what I want to do in this video is think about what is the equation of the tangent line when X is equal to 1 so we can visualize that so this is X equaling 1 right over here this is the value of the function when X is equal to 1 right over there and then the tangent line looks something like will look something like now I can do a better job in that it's going to look something like that and what we want to do is find the equation the equation of that line and if you are inspired I encourage you to be pause the video and try to work it out well the way that we could do this is if we find the derivative at x equals 1 the derivative is the slope of the tangent line and so we'll know the slope of the tangent line and we know that it contains that point and then we can use that to find the equation of the tangent line so let's actually just let's just so we want the equation of the tangent line when X is equal to 1 so let's just first of all to evaluate F of 1 so f of 1 is equal to 1 to the third power which is 1 minus 6 times 1 squared so it's just minus 6 and then plus 1 plus 1 minus 5 so this is equal to what 2 minus 11 which is equal to negative 9 and that looks about right that looks like about negative 9 right over there the scales are different on the Y and the x axis and so that is f of 1 it is negative 9 do I do that right this is negative 5 now you're signing up negative 9 and now let's evaluate what the derivative is at 1 so what is f prime of X F prime of X well here it's just a polynomial take the derivative of X to the 3rd well we apply the power rule we bring the three out front so you get 3x to the and then we go one less than three to get the second power then you have minus 6x squared so you bring the two times the six to get 12 so minus 12x to the well two minus one is one power so that's the same thing as 12 X and then plus the derivative of X is just one that's just going to be one and if you view this is X to the first power we're just bringing the one out front and decrement decrementing the one so it's one times X to the zero power which is just one and then the derivative of a constant here is just going to be zero so this is our derivative of F and if we want to evaluate it at 1 f prime of 1 is going to be 3 times 1 squared which is just 3 minus 12 times 1 so it's just minus 12 and then we have plus 1 so this is 3 minus 12 is negative 9 plus 1 is equal to negative 8 so we know the slope right over here is the slope of negative 8 we know a point on that line it contains the point 1 comma negative 9 so we could use that information to find the equation of the line the line just to remind ourselves has the form Y is equal to MX plus B where m is the slope so we know that Y is going to be equal to negative 8 X plus B and now we can substitute the x and y value that we know sits on that line to solve for B so we know that Y is equal to negative 9 let me just write this here Y is equal to negative 9 when X is equal to when X is equal to 1 and so we get we get negative 9 is equal to negative 8 times 1 so negative 8 plus B well let's see you could we could add we could add 8 to both sides and we get negative 1 is equal to B so we're done the equation of the line the equation of this line that we have in magenta right over there that is that is why is equal to the slope is negative 8x and then the y-intercept minus one