Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Use the first derivative to determine where the given function is increasing and decreasing.$h(x)=2 x^{2}-18$

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

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Oregon State University

Harvey Mudd College

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.

03:36

Use the first derivative t…

05:35

03:51

04:03

determine increasing and d…

03:23

02:02

02:59

00:55

Inspect the graph of the f…

05:01

04:06

04:13

00:56

Use the derivative to help…

01:24

Determine whether each fun…

01:03

Use the Monotonicity Theor…

This is Chapter three, Section two, Problem five. And this problem asks us to identify where this function each of X is increasing and where it is decreasing by finding the first derivative. So the first thing that we're going to do is find are critical points and the critical points if you remember our where the derivative of the function is equal to zero. So we're going to start off just by finding the derivative of H of X or H prime of X. And that's just going to be looks like four x so four X is only going to be equal to zero when X equals zero. So it looks like X equals zero is going to be our only critical point. Now that we've identified our critical point, we can make our number line. So right here in the center, our only critical point is going to be X equals zero, and we have zero for that value. So now we're going to choose a number that is less than zero and a number that is greater than zero for X. And we're going to plug that into H prime of X to see if it's going to be positive or negative and that will tell us whether our function is increasing or decreasing. So a number lower than zero we can just choose X to be negative one. And on the other side we can choose X to be one, and we'll plug those into h prime of x Mhm. So if we have a church prime of negative one that's gonna give us four times negative one or negative four because that sign is negative. That means that if we have X lower than zero, our function is going to be decreasing. Yeah. On the other hand, if we have a church prime of one, we get four times one, which is four. That's positive, which means if we have an X greater than zero, our function is going to be positive and increasing Mhm. So this means that for a given function h of X, it will be decreasing as long as X is less than zero and it will be increasing mhm right as long as X is greater than zero

View More Answers From This Book

Find Another Textbook

Numerade Educator

Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ be in any …

03:17

Approximate, using the method of the previous exercise, $f(31.99)$, if $f(x)…

02:42

Use the appropriate rules to determine the derivative.$$y=\frac{3 x^{5}-…

01:08

You are given a pair of supply and demand equations; identify which is suppl…

03:00

Determine the equation of the tangent line at the indicated $x$ -value.$…

01:22

Given the data set $(-1.1,-9.1),(0.05,-9.02),(1.05,-10.9),(1.95,14.9) (2.9, …

08:06

(a) John is at $B$, on a straight beach, 10 miles from $A$. Mary is in a boa…

01:49

Locate all critical points.$$f(x)=\frac{x+3}{x-3}$$

Use the first derivative to determine where the given function is increasing…

02:29

Find $d y / d x$ if $y=\sqrt{\frac{x+1}{x-1}}$.