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 the points on the curve $$y=2 x^{3}-54 x+1$$ at which the tangent line is horizontal.

$$y=\frac{7}{2} x+1$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Campbell University

Baylor University

University of Nottingham

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.

06:02

Find the points on the cur…

01:28

02:15

Find the coordinates of th…

00:49

Find the $x$-coordinates o…

Find the point(s) on the g…

02:36

05:08

Find all $x$-coordinates o…

The underlying theme in this is if you have a horizontal tangent that basically we're saying I want to use the same notation because they're talking about why equals then d y d x at certain X values may Don't even say it x equal c I must have the equation of the derivative should equal zero somewhere. And, uh, usually there's a little space there. Sorry about that. Um, that's the overall premise of this problem. So what we need to do is personal right out the equation two x cubed. But then what I'm gonna dio okay is find the derivative of this. So do you wind the X? What, equal six? Because you bring them three in front times to a six x squared minus 54. We're going to set that equal to zero. So now what I can do is add 54 over divide by six to solve for X squared. Um, but I would simplify that because both those numbers are divisible by Ah, I think by six, right, you will get nine eso when you square root both pieces. Um, you would get plus or minus three. So the two places on the point on the curve. Uh, now, what I need to do is go back to the original problem and plug in three and negative three. And for these exes to figure out what those values are. So three Cube is 27 times two of the 54 minus 54 times three plus one, which should be 54 times to be negative. 108 plus one would be negative. 107. So that would be a point if excess positive doing this in my head, by the way. So if I'm slightly off, I'm sorry on then, Negative three. When I cubit will be negative 27 times to be a negative 54 and there will be a plus a 54 times three. So that should be a positive 108 plus one would be 109. So, assuming I did my math correctly, I think other eight points here

View More Answers From This Book

Find Another Textbook

Numerade Educator

00:43

Short segments of the tangent lines are given at various points along a curv…

00:56

Sketch the graph of a continuous function defined on $-5 \leq x \leq 3$, tha…

01:22

Suppose the demand equation is $200 x+10 p=10000,$ where $x$ is the number o…

02:53

Sometimes when one plots a set of data, it appears to be piece wise linear, …

01:06

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

01:18

Given the graph of the function $y=f(x)$ in Figure $30,$ draw the graph with…

02:52

Find $\frac{d}{d x}\left(\frac{4 x^{6}+3 x^{3}-8 x}{6 x^{5}}\right)$ by: (a)…

04:13

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

05:48

(a) Determine the $x$ -intercepts, (b) the vertical asymptotes, (c) the hori…

03:34

Given the data set $(1,0.9),(2,3.8),(3,6.2),(4,4.1),(5,2.2),(6,5.8),(7,9.3)$…