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Find the average rate of change of $y$ with respect to $x$ on the given interval.$$y=2 x^{2}-7 x+2 ;[2,3]$$

$$3$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

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So to find the average rate of change between points two and three, we can use the formula. Why, too, minus y one over x two minus X one. Because this is our average rate of change function. So to get y two and y one first, here's our X one and X two. So to get our y two and y one, you plug in X one to the function and get an answer, and then x two into the function and get an answer. So if we plug in X one and you just plug and chug, which is plugging into to to where the X valuables are in solving that you get ry one is equal to negative for. And then if you plug in your X two we get are y two equal to negative one. So if we put this these two back into our function, we get negative one minus negative floor over three minus two, which equals three. And what that means is, if you have a quadratic equation, let's just say that's a very generic quadratic equation. Not this one. Let's say points two and three are here. So for this part of the function, the average rate of change is three units. Mm

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