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Let $y=a x^{2}+b x+c .$ Find the slope of the chord (see Exercises 34 and 35 ) connecting the points with $x$ -coordinates $r$ and $s$. Find the $x$ -coordinate on the parabola where the tangent line to the curve has the same slope as the chord.

$$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \mathrm{h} & -.1 & -.01 & -.001 & -.0001 & .0001 & .001 & .01 & .1 \\\hline m_{P Q} & 4.6416 & 21.544 & 100.00 & 464.16 & 464.16 & 100.00 & 21.544 & 4.6416 \\m_{P Q} & -0.4642 & -0.2154 & -0.1 & -.0464 & .0464 & 0.1 & 0.2154 & 0.4642 \\\hline\end{array}$$in $f(x)=x^{1 / 3},$ the slope is becoming infinite as $h$ approaches zero for $f(x)=$ $x^{4 / 3}$ it approaches 0

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:11

Given the parabola, $y=x^{…

01:49

01:15

Find a point on the curve …

01:34

Let $E_{l} E_{2}$ and $F_{…

05:01

(a) Find the slope of the …

01:31

The locus of the mid-point…

01:25

eso This seems like a goofy problem to me, but that's okay. We can go ahead and do this. Um, And what they're telling you is that we want X coordinates. Where, uh, one x coordinate is our So I'm talking about the point. Then it would be plugging are in for X and the other point being s my s is kind of look like five. So I don't think I have five written anywhere on the page. So if you see a and that s that's what this So the first point is they want you to find the slope between those two values which I'm going to do you know why? To minus y one. So I'm looking at a s squared plus B s plus C, and I'm gonna subtract off the wide coordinate now notice I am distributing. All of those over s minus are so we can simplify. You know, for instance, um sees cancel, um, and then of everything else I could if I wanted to. Um, like, factor out in a and do s squared minus R squared and then plus factor out of B and B s minus are and in the denominator you have s minus are still so that's still equal to the slope. Hopefully all of that made sense. If you don't believe me, you start distributing, you know, see, you will be equal The a s squared my minus a r squared plus B s minus b r But the reason why I'm showing you this step is because then we have a difference of squares there in the numerator Um, where we can write a times s plus are s minus are plus B as minus are And because all of these terms share and s minus are weaken Divide each piece out So the slope between any two points are and s alright, it over here and you could distribute if you want a s plus a are, uh plus B So what they're trying to do is ask you then. Well, what point for X would give me that same slope? Well, you can figure that out by finding the derivative of this, So I'm gonna do that down here. So we keep this in the back of my mind so d y d X is gonna equal to a X plus B and the derivative of a constant is zero. And what I need to do is set that equal to the slope A s, plus a r plus B, the slope that we found and the first thing you should notice There, I hope you notice, is that these bees will canceled and we can divide out. I mean, technically, I'm factoring out and a and all of this I'm looking at two X is equal to a s plus are. But the whole point of that is that I can cancel out these ays, um, we divide them each. Their constant, I guess, would be another way of saying this. So to solve for X, I can divide that to over. So the whole premise of this problem is that the X coordinate on a parabola that is, um, parallel to the secret line between any two points is the halfway. So X is halfway. I hesitate to say midpoint because some students confused the word midpoint. Ah is halfway between r n s so kind of interesting. Anyway, there you have it

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04:10

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