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Use the first derivative to determine where the given function is increasing and decreasing.$s(t)=t^{2}+4 t-21$

Dec on $t < -2 ;$ inc on $t > -2$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

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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.

02:59

Use the first derivative t…

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01:02

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04:06

This is chapter three, Section two, Problem six. In this problem, we are asked to find where this function s of t is increasing in where it is decreasing by finding the first derivative. So what we're going to do first is find the critical points. And if you remember, the critical points are where the derivative of the given function is equal to zero. So that's going to be where s prime of T is equal to zero. So let's first find that derivative s prime of T that's going to be equal to to t plus four. Yeah, Yes. Now, to find the critical points, we can factor this out. We can factor out the to to get T plus two that's equal to the same thing. So now we know that this function can only be zero. If this is zero in t plus two can only be zero If t is equal to negative two. So t equals negative two. Oops. That's going to be our critical point. So now we can draw our number line. Yeah. Here we have t equals negative two. If we plug that in, we get zero down here will choose a number that is less than negative two. So let's just say negative three. In here, we'll put a number greater than negative two. So let's just put zero. And we're going to plug both of those numbers into s prime of T to find whether our function is increasing or decreasing. So if we plug in negative three for tea, we'll get to negative three plus two, and that's going to be equal to negative two because it's negative. That means that our function is decreasing when t is less than negative two on the other side, if we plug in zero, okay, we get four, which is positive. And that means that if we have a t that is greater than negative to our function is going to be increasing. Yeah. So for our function s f t, it will be decreasing. Yeah, if t is less than negative two. Okay. And it will be increasing if t is greater than negative two. And that is your answer

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