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Use the first derivative to determine where the given function is increasing and decreasing.$f(x)=m x+b,$ if:(a) $m>0$(b) $m<0$

(a) always increasing(b) always decreasing

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

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University of Michigan - Ann Arbor

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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This is Chapter three, Section two problem, too. And in this problem, you're asked to use the first derivative to find whether this function f of X, where it is increasing and where it is decreasing. If A M is greater than zero and B F m is less than zero. So the first thing that we're going to do is find are critical points, and this is where the derivative, our function is equal to zero. So before we get started with our EMS A and B, let's just first find the derivative of this function. F a bex. If em is going to be a number, then we're going to solve this like how we would any derivative and keep that end there. And then the X goes away and the B goes away because it's a derivative of constant, and that's going to be zero. So our F Prime of X is just going to be equal to M. So if we look at a M is greater than zero, let's it can be any number greater than zero. So let's just say I'm is equal 21 That means our f prime the Becks is going to be equal to one. It's just going to be equal to whatever M is because it's positive and because it's a positive constant. That means that no matter what we plug in for X here, we're always going to have a positive number here, which means for M greater than zero, the function is always going to be increasing. You won't. The same applies for M less than zero, so we can do the same thing here. M equals negative one. So we have f prime of X equals negative one. So again, this is always going to be a negative constant, which means no matter what we have for X, we're always going to have this negative sign here, meaning that the function is always going to be decreasing, and that's your answer for this problem.

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