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consider the function f of X is equal to one over x squared minus KX where K is a nonzero constant the derivative of F is given by and they give us this expression right over here it's that's nice that they took the derivative for us now Part A let K equals 3 so that f of X is equal to 1 over x squared minus 3x so they set k equal to 3 write an equation for the line tangent to the graph of F at the point whose x-coordinate is 4 so to find an equation for line the equation of a line is going to be of the form y is equal to MX plus b where m is the slope of the line and B is the y-intercept and the slope of the line right over here this is this needs to be equal to the derivative evaluated when X is equal to 4 so we could say Y is equal to let me write it this way we could say that M is going to be equal to F prime when X is equal to 4 so F prime of 4 which is equal to well we know that K is equal to 3 so they gave us F prime of X so it's going to be 3 minus 2 times we're taking F prime of 4 so minus 2 times 4 over x squared so that's going to be 4 squared minus K is 3 times 4 3 times 4 and then we square that whole thing and so what is this going to be this is an 8 right over here and all I did is F prime of X when K is equal to 3 is going to be 3 minus 2x over x squared minus 3x and all of that squared and I want to evaluate what F prime of 4 is so every place where I saw an X I substitute it with the 4 where I saw the K K is 3 and so this is going to be equal to the numerator is 3 minus 8 is negative 5 over this is 16 minus 12 which is going to be 4 16 minus 12 is 4 and then we square it so it's going to be negative 5/16 and so let me write this way M is equal to negative 5 60 so how do we figure out B now well what are the coordinates when X is equal to four what is y going to be equal to well Y is equal to f of X so we know that Y on the curve we know that Y is going to be equal to F of four so before we evaluate F prime before now we're going to evaluate Y as being F of four which is equal to 1 over 4 squared 4 squared minus three times four and so that is equal to 1 over 16 minus 12 which is 4 and so this point right here when X is 4 then Y is equal to 1/4 so we can use that information to solve for B when when y is 1/4 X so we're going to say if Y is equal to M which is negative 5 16 negative 5 16 times X well when y is 1/4 X is X or when I say when X is equal to 4 y is 1/4 and then plus B so I can now solve for B so all I did is I used F prime of X I used F prime of X to figure out M when X is equal to 4 and then I said ok well what is the value of y when x is equal to 4 and so if I know Y M and X then I can solve for B and so let's just do that we get 1/4 is equal to 4 times negative 5/16 is negative 5 over 4 plus B I can add 5/4 to both sides and I get 5/4 plus 1/4 is equal to B or B is equal to B is equal to 6 fourths 6 over 4 which you could say well there's a bunch of ways you could write this we could just say this is equal to 1.5 and so our equation is y is equal to negative 5 over 16x plus 1.5 or if we wanted to write everything as a fraction we could say Y is equal to negative 5 over 16x plus three-halves 6/4 is the same thing as three as three has and there you go

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