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Find the average velocity over the given time interval, if $s=f(t)$ is the equation of height as a function of time.$$s=-16 t^{2}+128 t:(a)[0,3] ;(b)[3,4] ;(c)[0,4]$$

(a) 80(b) 16(c) 64

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Campbell University

University of Nottingham

Boston College

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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to find the average rate of change. You basically use the equation Change in US corporate change in t. So for each of these points we solve for each of the variables. So here's us one. Then here's s or sorry, t one and T two right here. So after yourself for T one and t two, you get s one and s two as your results. So to do a for S of t one where basically you plug in this value zero into each of the t terms, you get zero. And then for t two, you plug in three into each of the T terms. So negative 16 times three squared plus 1, 28 times three you get 80 or sorry, 240. So after you do this, you have your s one and s two. So this is s one. This is s two. So you can plug these values into the equation up here, which is basically as to minus s one over t two minus t one. What you get? Yeah, Yeah. 2. 40 over three, which is equal to 80. So that's your answer for part A for part B you do the same thing you plug in three into your T variables to get s one, which will give you we already did it above 2. 40. And then you plug in four into each of your T variables to get s two for the point of four, which gives you 256. So then you can plug in charge. You get to 56 minus 2. 40. What over four minus three, which is equal to 16. So that's your average rate of change between the points of three and four. Finally, our last point, we have zero and four, so we already did calculate the aspect values for each of these points. S F T for zero is equal to zero. That's your s one and s of T 44 is 256. So that gives you 2 56. So the average would be to 56 minus zero over the points of four minus zero, which gives you 64 says your average rate of change over the points over the range of 0 to 4

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