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 and second derivatives to sketch the graph of the given equation. Also include the intercepts, whenever they are easily determined.$$f(x)=3 x /\left(4 x^{2}+9\right)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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.

03:01

Use the first and second d…

03:05

02:57

03:14

02:48

01:11

01:44

02:30

01:23

03:33

03:32

03:07

02:02

01:21

03:41

01:06

02:44

02:53

09:56

01:12

we went to graph out of X equals three X over four X squared plus nine. To do so we're gonna deploy calculus curve sketching techniques as follows. First We analyze after basic info F. Gives us domain all X Y intercept X equals zero Horizontal assistant white was 3/4 at exclusive responsibility and no vertical absent votes. We need to analyze F prime for information about where F is increasing or decreasing. F prime is negative three times for experiments 9/4 X squared plus nine squared. There's petitioner critical points at plus or minus three halves. So we evaluate on these intervals F prime to the left of negative three half I'm is negative so decreasing for F between positives are increasing To the right of 3/2, negatives are decreasing again. Finally we analyzed double prime from cavity purposes. That's the problem is 24 X times for experiments. 27/4 X squared plus nine cubed. That's why the petitioner critical points at zero plus or minus three or three over to the US. We have assigned a vegetable prime for X less than negative. 3, 3/2 negative. So if they can keep down between negative three or 3/2 and zero double prime is positive, so can keep up, then congregate down concrete up again. Therefore, we can grasp. We have our noted changes from decreasing the increasing, increasing, decreasing it negative three hats and three has marked. We also have a change in con cavity displayed to the left, negative three hats and to the right of three hats.

View More Answers From This Book

Find Another Textbook

02:38

Determine whether or not the function determined by the given equation is on…

03:50

A machine worth $10,000$ new and having a scrap value of $500$ is to be depr…

03:06

$\$ 4750 dollars is deposited into an account for 20 years. Determine the ac…

01:32

Sketch the graph of the function defined by the given equation.$$y=f(x)=…

00:50

Determine if the equation describes an increasing or decreasing function or …

Use the first and second derivatives to sketch the graph of the given equati…

02:37

01:25

Write the given expression as a sum of logarithms.$$\log _{3} \frac{2}{y…

01:05

The line $y=m x+b$ is called a slant or oblique linear asymptote of the func…