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(a) Find the slope of the tangent line to the parabola $ y = 4x - x^2 $ at the point $ (1, 3) $

(i) using Definition 1 (ii) using Equation 2

(b) Find an equation of the tangent line in part (a).

(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point

$ (1, 3) $ until the parabola and the tangent line are indistinguishable.

a) $m=2$

b) $y=2 x+1$

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Campbell University

Oregon State University

Idaho State University

So we're asked to find the slope of the tangent line at a problem. Considering the equation of the problem is four x minus x squared. And this this tangent line is going to be at the 0.1 comma three. So first of all we're asked to find the slope of this tangent line. Well the slope is the derivative dy over dx which is Taking the derivative that that's 4 -2 x mm. So at X equals one. Then the slope is four minus two. Which is to. Then we're asked to find the equation of this tangent line. We know by using the point slope formula because we have both a point on the line and slope of the line. We know that why minus why one is equal to m times x minus x one. So this is why -3 is equal to two times X -1 which gives us y equals two X two times -1 is -2. And then we're gonna add three to that to get that to the both sides, that's two, X plus one. Then we're asked to graph all of this. So let's get our graphing up here And a graph of our Parabola 1st is for x minus X. Brand, is there a problem? And then the equation of our line is too X lost one until therefore we now have uh huh We now have the quake they had problem graft here and the equation of our tangent line. And notice the intersect right there at the 0.1 comma three

Oklahoma State University