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

Sketch some of the level curves (contours) or the surface defined by $f$.$$f(x, y)=x^{2}-2 x-y$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

04:06

Sketch some of the level c…

04:42

09:37

12:10

01:14

Sketch some level surfaces…

00:44

00:43

00:37

So if we want to draw some of the contours for this surface here, remember, what we're going to do is just say this is equal to some constant Z, and then we can just try to graft this with different values of Z. So the first thing I'm gonna do is just kind of move things around in salt for why? So I'd add Why subtract Z So this is going to be why is equal to X word minus two X, um and then minus Z here and now an easy way for us to kind of graph this for all values. Um, just gonna do it quickly is to go ahead and complete the square of this. Um, so that's just what I'm going to do really quickly. So remember to complete this square here we take the middle term divided by two square and then add and subtract s O that would end up being X minus one squared, and then I'd have to do plus one minus C. And the reason why I want to do this here eyes because are actually let me keep this blue here, Um, find a C S o the reason why I want to do this is since this is a quadratic graphing it in This Vertex form is a lot easier than pretty much using anything else. Because at this point, we could just, like, shifted up and down by some value. Um, yeah, that's the only reason why I really wanted to do that. So let's come over here now, So let's first do. Like, um, Z is equal to zero. So if I do is easy to zero that I end up with. Why is it you two x minus one squared plus one. And so remember the vertex of this. Actually, if I were to look at it here, the vertex of this would be at one and then one minus Z. So going positions that's really going to be changing is easy because we're gonna be shifting this up and down by ah factor of Z or not by a factor by just a Z units. Um, so let's put some of these towns of 1234512345 MLB five going up as well. Five going down. All right, so this first one is going to have the Vertex at 11 So right here. Um then I could just put a couple of other places. And so I believe this is what this is going toe look like. So, yeah, we've just graph that so that Z is good zero. We normally write off on the side. What the level curve is for this. Now, if we were to do like, Z is equal to negative one or a very old it was easy with one first that's going to give us why is equal to, um So we just plug in negative one or one here, so there will be one minus one, which would just be X minus one squared. And so all that really does is essentially just shift it down one unit. So it would be like this here, and this is going to be Z is equal to one. Um, let me go ahead and do like Z is equal to. So if I were to plug to into here, that would give us X minus one squared minus one. And so again, that's just going to shift this town. One more unit. So be here. Here, here. And it was a little bit too far down. I think it should have been like there. And then this is going to be our level curve for a Z is equal to And then let's just do Ah, negative one. So we do like ZZ put a negative one. So instead of subtracting now it's socially like I'm adding causes beat negative. Negative one. So it be X minus one squared plus two. So again, from that green one, it just shifts it up. One unit, +111 But then here and here. And if you want, you could go ahead and do some Maurer of the level curves. But I think this is a good place to kind of something. I don't know why I put negative here. Are actually guess those air my equals just draw them too Well, yeah. So those were just some of the level curves. Um, you could doom or if you want, but I think at least doing these four is a good enough

View More Answers From This Book

Find Another Textbook

02:08

Sketch the area represented by the given definite integral.$$\int_{-1}^{…

01:32

Evaluate the given integral.$$\int \frac{e^{2 x}}{e^{2 x}+1} d x$$

01:42

Evaluate the given integral and check your answer.$$\int \sqrt[4]{t} d t…

02:46

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject…

02:29

Evaluate the given definite integral.$$\int_{0}^{\sqrt{15}} \frac{x}{\le…

01:52

(a) Compute $\frac{d}{d x}(x \ln x-x)$. (b) What function can you now antidi…

02:10

Sketch the area represented by the given definite integral.Integral in E…

01:02

Solve the given differential equation.$$\frac{d y}{d x}=3 x^{4} y^{4}$$<…

01:56

Evaluate the given integral and check your answer.$$\int\left(5 x^{2}+2 …

02:18

Find and classify, using the second partial derivative test, the critical po…