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(a) Find the instantaneous rate of change of $s$ as a function of t. (b) Find the value(s) of $t$ for which the instantaneous rate of change is positive.$$s(t)=3 t^{2}-12 t$$

(a) $6 t-12$(b) $t>2$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

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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let's suppose that F F T. Equals t squared. Yeah. Mhm. Um Plus three p -7. Right. One of the average rate of change over the interval from 5 to 6. So it's gonna be FF six -F of five. And then we're going to divide that by 6 -5 For the 14. But then we can find the instantaneous rate of change. Um and assuming that we're looking at five, we'll bring this in closer, We see the desire that's being approached, We'll be 13th, lets the slope we're looking for. And if we change this to be three T plus two, yeah mm minus hover over key. And we want to find the average rate of change Over that interval. We want to know what the instantaneous rate of change is when T equals two. Um So we'll have three here and true here. So the average rate of change is going to be even process five. So as we get closer of 2.001 for example, And we see that with this problem, six is going to be the slope of being approached

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