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Given the parabola, $y=x^{2}+3 x-5 .$ (a) Find the slope of the line connecting the points (1,-1) and (3,13) on the curve. (b) Find the point on the curve at which the slope of the tangent line is the same as the slope of the chord.

(-4,-64) and (2,8)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

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.

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Given the parabola, $y=x^{…

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Let $y=a x^{2}+b x+c .$ Fi…

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Consider the parabola Y = …

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Determine the point on the…

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Find the parabola with equ…

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(a) Find the slope of the …

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Determine the slope of the…

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Find the slope of the tang…

So what we need to do in this problem is look at the ordered pairs. One negative one and through 13. And if you're confused where these ordered pairs come from, if you were to plug in one and for all these exes, one plus three is four minus five is negative one. Um, same thing if you plug in three. And for these excess you get nine plus nine 18 minus five is 13 eso. The first part of this problem is just examining if you find the slope, which is the ordered Sorry, the formula y tu minus y one over x two minus x one. And as you simplify, you get a slope of seven for that cord. So that's the first part of this problem. So then the second part is can you identify the point on the curve that whose slope is equal while you find that by taking the derivative, which is our rules of two X plus three and you want to set that equal to the soapy found in part a, which as you subtract three over, uh, you get seven minus three is four and divide by two. You get X equals two. Now it does ask for the ordered pair eso the point so you don't want to plug in to and from both the sexes. So that would be four plus six is 10 minus five will be five. So the ordered pair is 25 Um, and so the only other thing to mention I don't know if I even need to is that there's a reason when you're looking at a quadratic, that if you pick any two points, the tangent line that has the same slope will be halfway between the two points of that cord. So halfway between one and three is X equals two, and that is proven.

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