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

Find those points at which the tangent line to $y=f(x)$ is horizontal for. (a) $f(x)=x /\left(x^{2}+1\right)$

(a) (-1,-2),(1,2)(b) $(-1,-1 / 2),(1,1 / 2)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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.

04:17

Let $f(x)=\frac{x}{x^{2}+1…

04:53

Horizontal Tangent Line De…

05:11

Let $f(x)=\left(x^{2}+1\ri…

All right, so we're looking at X over X squared plus one versus X Squared plus one over X. And any time we have horizontal tangents. Basically, what it's saying is we want to take the derivative, however notation they use it like like in this problem, they use ffx. Um, so you know, we're looking at f prime of X needs t equals zero. So as we examine the first one male do this first one in red. Uh, right here. Well, the question will tell us the derivative of the top is one. You leave the bottom alone and then minus the derivative of the bottom, you leave the top of them all over the denominator square. Now, we really don't care about the denominator, especially in this case, because the denominator can never equal zero. We only care about the numerator. So as I just examined the numerator, we have X squared plus one, uh, minus two x where X squared minus to export to give me negative one x squared plus one. I want to say that piece equals zero. We really ignore the denominator. Because if the denominators hero just undefined, we only care about the new Marie. That's the only way to get this equal to zero. So as we solve that export illegal one. So exc unequal plus or minus one as we go back to the original problem. If X is one that we get one half of the Y coordinate 11 half and effects is negative. One, uh, next one divided by two. It's still one half, isn't it? I think one would happen. This is negative one. Yeah, negative One half way Go. Because if we look at the other one now, look at this one and we do the same thing. That drift of the top is two x Really. The bottom alone, minus the group of bottom lead the top alone. What you'll see is all over the denominator squared. But again, it only equals zero of the numerator. Call zero's air looking at one X squared minus one because you've got to distribute that equals zero notice. What happens, though, is you add one over and divide you get plus or minus one again. But now you're ordered pair know anything that's different is you know, one. Um um what do you get? One squared plus one is 2/1, and then the other ordered pair of the negative one squared is still to divide by negative one, though you get negative too, so you should notice is the export. It's really the same for Parts A and B, but the only thing that's different is the Y. Coordinate. So these are your two answers for A and B.

View More Answers From This Book

Find Another Textbook

Numerade Educator

03:04

Classify all critical points.$h(x)=\left(12-x^{2}\right)^{1 / 2}$

02:33

Find $d y / d x$ using any method.$$x^{2}-x y+7=0$$

01:16

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

01:29

Determine where the function is concave upward and downward, and list all in…

03:09

Consider the farmer in Exercise 7 whose field borders the river. Assume he w…

01:39

Find the average rate of change of $y$ with respect to $x$ on the given inte…

05:31

A rubber ball (a sphere) is expanding in such a way that its radius increase…

03:24

Find the average velocity over the given time interval, if $s=f(t)$ is the e…

02:15

Consider the function given by$$f(x)=\left\{\begin{aligned}-2 x+1 &a…

05:48

(a) Find $d$ fif $f(x)=\frac{2 x+3}{x^{2}-2}$,(b) Find $d y$ if $y=\frac…