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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.

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a) $m=2$b) $y=2 x+1$

06:46

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

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University of Michigan - Ann Arbor

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Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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

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