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Use the first derivative to determine where the given function is increasing and decreasing.$g(t)=t^{2}-1$

Dec on $t < 0 ;$ inc on $t > 0$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Harvey Mudd College

Baylor 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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This is chapter three, Section two, problem for And this problem is asking to identify where this function GFT is increasing and where it is decreasing using the first derivative. So what we're going to do is find are critical points first, and that's going to be where the derivative of this function G prime of T, is going to be equal to zero. So let's first find the derivative of GFT, which is going to be g prime of T. And that's just going to be equal to to t. Okay, so the critical point is where G prime of T is equal to zero. So to t is only going to equal zero when t is zero. So already we know we have a critical point of T equals zero. From here, we can create our number line, okay, with T equals zero right here in the center. And we know if we plug that in, we get zero. So now we can plug in anything T that is less than zero. And on our other side, we're going to have a number that is greater than zero, and we're going to see if our function is increasing or decreasing. Mhm. So, just for simplicity, our number that's less than zero can just be negative one. And our number greater than zero can be one. And we're going to plug those both in two g prime of T and see if it's positive or negative. Okay, If we plug in negative one into our function, we're going to get two times negative one. That's just going to be negative, too. So it looks like if we have t less than zero, we're going to have a decreasing function. Yeah, and if we plug in one, we get just the opposite, which is to so t greater than zero is going to be positive. This means that for this function GFT mm. Evidence three. So, in conclusion for our function, G of T, it will be decreasing. Okay. When t is less than zero and it will be increasing Okay. When t is greater than zero

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