(a) Find the slope of the tangent line to the parabola y=4...

(a) Find the slope of the tangent line to the parabola $y=4 x-x^{2}$ at the point $(1,3)$ (i) using Definition 1 (b) Find an equation of the tangent line in part (a) (c) Graph the parabola and the tangent line. As a check o your work, zoom in toward the point $(1,3)$ until the parabola and the tangent line are indistinguishable.

(a) Find the slope of the tangent line to the parabola
$y=4 x-x^{2}$ at the point $(1,3)$
(i) using Definition 1
(b) Find an equation of the tangent line in part (a)
(c) Graph the parabola and the tangent line. As a check o
your work, zoom in toward the point $(1,3)$ until the
parabola and the tangent line are indistinguishable.

Key Concepts

Derivative

The derivative of a function at a given point represents the instantaneous rate of change of the function with respect to its variable, or geometrically, the slope of the tangent line to the graph of the function at that point.

Limit Definition of Derivative

This concept defines the derivative as the limit of the difference quotient as the increment approaches zero. It provides a fundamental method to compute the slope of the tangent line by evaluating the limit of (f(x + h) - f(x))/h as h tends to 0.

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve at that point and has the same instantaneous slope as the curve at that point. It is an important concept in calculus for approximating the curve locally.

Point-Slope Form of a Line

This is a formula used to write the equation of a line when the slope and a single point on the line are known. It is typically expressed as y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.

Transcript

00:01 In this problem, we are asked to find the slope of the tangent line to a given parabola at the point 1 3.

00:07 We can find the slope of the tangent line using either definition 1 or equation 2.

00:12 Here we see that we have definition 1, where we can say that the slope of our tangent line is equal to the limit as x approaches a of our original function minus our function evaluated at sum value a, divided by x minus some value a.

00:28 And here we define a to be the x value at the desired point where we are drawing our tangent line.

00:34 So in this case, we can say that a is equal to 1.

00:39 So here we have definition 1, where we plugged in the number 1 for a.

00:43 We plugged in our original function, and then we plugged in our function evaluated at the value of a.

00:49 So we can start to simplify, first simplifying our numerator.

00:55 And when we do that, we end up with the following polynomial divided by x minus 1.

01:06 And we can notice that we can simplify and factor this polynomial that we have on top.

01:14 So when we do that, we end up getting x minus 1 times 3 minus x up top, and that is divided by x minus 1.

01:24 So we can go ahead and cancel these two quantities, and then we end up getting that we have the limit as x approaches 1 of 3 minus x.

01:37 So when we go ahead and evaluate our limit here, we get that the slope is equal to 3 minus 1 or that the slope is equal to 2.

01:47 We can go ahead and repeat this using equation 2.

01:51 With equation 2, we have a similar approach for finding our slope where we're using a limit where h is approaching 0.

01:58 We see that we have our function, and we're plugging in a plus h for any value of x, and we're subtracting our original function evaluated at the value of a, and this is all divided by the variable h.

02:13 So in this case, similar to our definition 1, a is also going to be equal to 1, because it is also defined as the x value at the desired point where we're drawing our tangent line.

02:25 So just as we did with our definition one, we can go ahead and start simplifying.

02:31 Here we have our limit where we've plugged in all of the important values.

02:36 We have our function where we've plugged in a plus h.

02:40 And then we have our function evaluated at a.

02:43 And we found that this was equal to three above by simplifying what we have in the parentheses here.

02:49 So we can go ahead and begin to simplify by distributing and simplifying what we have in our numerator once again...

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