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.$$f(x)=x-x^{2 / 3}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Baylor University

University of Nottingham

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.

03:05

Use the first and second d…

03:14

03:01

02:02

03:07

01:23

01:30

02:48

03:32

03:33

03:41

01:44

02:30

01:11

01:06

01:21

02:53

02:44

09:56

01:12

question 63 wants to use the 1st and 2nd derivative to sketch the graph of f of X equals X minus X to the two thirds, uh, and then use any intercepts to help you do so if it's easy to find, um, what intercept you can find right away is when why is equal to zero. So setting this function equal to zero is when x is zero and also one. So those are two points you can grab, then looking for critical points at prime of X is equal to one minus two thirds X to the negative one third simplified one minus two over three X to the one third. Yeah, uh, now looking for when this is equal to zero for our critical points, you would have to over three x to the one third equals one to over three equals X to the one third. Therefore, X is equal to 8/27. And from there you also can say that, uh, this function is undefined at X equals zero. Therefore, when we're looking at the con cavity, we can test around that point. Um f of 8/27 is equal to negative 0.1 for eight. So there's another point to graph uh, now looking at Con Cavity, we need F double prime of X, which is just to over nine x to the negative 4/3, which is equal to 2/9 to the X extra four thirds from their testing when X is less than zero. So at double prime of negative one is 0.2 to your concave up in that region, then looking at when not only is x greater than zero, but are critical point here of 8/27 after a prime of 8/27 is 1.125 So you are also concave up. Um, using all this information, we can go ahead and plot our points on a graph. So we have 00 and we have 10 then our other point is 8/27 about here and negative point 148 So about here, you know your concave up before zero. So here, down to the minimum and back up. And that is what your graph looks like for question 63

View More Answers From This Book

Find Another Textbook

03:55

Find the equation of the tangent line to the curve at the given $x$ -value.<…

03:03

Sketch the graph of the line $x-2 y+1=0$ on the same axes as the graph of $y…

04:20

A 26 foot ladder is leaning against a vertical wall and its base is on level…

02:03

Explain the relationship between the answers in parts (a) and (b) of Exercis…

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

04:24

(a) Sketch the graph of the line $y=2 x-3$ on the same axes as the graph of …

02:14

Determine the equation of the inverse function.$$f(x)=2 x^{2}+3, x \geq …

01:10

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

03:25