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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.$$s=-16 t^{2}+160 t:(\mathrm{a})[0,3] ;(\mathrm{b})[3,6] ;(\mathrm{c})[3,10]$$

(a) 112(b) 16(c) -48

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

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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01:17

to find the average rate of change between the points in the question. We basically use the equation, the change in s or change in T, which is also as to and his s one whatever to you too. And it's t one. So if you look at our points here, we have our T ones and t two's here. So 03 and three R T ones and 36 and 10 or T tubes. So to find us to an s one, you basically plug and chug each of the very Each of these points into our equation for tea. So if we look at part A, the T 20 equals zero s equals zero. So that is our t r r s one and R T one for part a. And then if you look at 3 20 equals three yeah, s equals and you plug you plug and chug three for the T and you get as equals 336 so that as our t two and s two. So to find the average rate of change for between the points zero and three, you plug these four variables into our equation up here so as to is 3 36 minus s one, which is zero over t two, which is three and t one, which is zero. And that gives you 112 as the average rate of change or the average velocity between those two points. So for part B, we did the same thing when t equals three. We already solved up here as equals 336. So that gives our t one n s one for this. This this set of points and then when t it was sex, we plug six in two R T variables up here. Uh huh. And we get s is equal to 384. So that gives our t two and R s two for this set of points. So then we plug that in, I'll go back to Green here to our equation as to minus s one over t two minutes, t one and we get 3 84 floor minus 3, 36 over six, minus three. And that gives us 16 as the average rate of change between the points three and six. So we're going to stick with green here. So now we go to the last set of points, which is t one equals three, which we already solved. That will give us Rs One is 3. 36 mhm. And then our t two s 10. You plug in 10 to the T variable in the equation and you get when t is 10 as equals zero. So you plug that in to the final, the rate of change equation or the average velocity equation. You get 3 36 minus zero running out of space here over 10 minus three. And that gives you a rate of change of I was sorry. It would be zero minus 3 36 over 10 minus three, Mhm zero minus 3. 36 over time. And that will give you a real change of negative 48. So, basically, we'll put this in on the top here. If you have a quadratic in this case, a negative quadratic between this point between zero and let's say a point here, an arbitrary point, your rate of change is positive. And then at the top, when you're going back down, it is negative. So, between this point at this point, yeah,

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