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Use the guidelines of this section to sketch the curve.$$y=x^{3}+3 x^{2}$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 5

Summary of Curve Sketching

Derivatives

Differentiation

Volume

Missouri State University

Campbell University

Harvey Mudd College

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we want to sketch a graph of why is he with that excuse? Plus three x squared. So in the shocker against us, a laundry list of how we should go about doing this. So Step one was first to identify our domain. And we know polynomial is have a domain of all real numbers, just negative entity to infinity. Then it suggests we move on to our intercepts. So let's go ahead and start with the ex intercepts. So ex intercepts. Well, this means why is equal to zero. So zero is equal to x cubed plus three x squared and noticed We compact out of X squared. Thanks, Bus three. And this tells us X is equal to zero or ex busy, too. Negative. Three. So we have our duplex values or X intercept. No, let's go ahead and find out why. Intercept. Remember, this is we set X equal to zero, so you might be able to notice that from movie did in the last part, we should be able to tell, but we could just go out and plug this end. So zero cubes zero and then zero squared to zero. So we get our wide value of zero for the wide, interesting. Next. It wants us to check for any kind of symmetry. So we want to figure out the functions, even or odd or anything like that. So let's go ahead and put a little biter here. So in the tree. So if this year is supposed to be EPA Becks, let's go ahead and look at what of negative exes. There's gonna be negative x cute. What's three negative X squared, which is going to be negative, X cubed, plus pretty X squared. So since this does not equal to EPA, Becks or negative at the vex, we have no symmetry. Next, it wants us to find acid trips. So if we take the limit so the minute as X goes to infinity of f of X. Well, since this is a polynomial, we know that this is going to go to infinity and the limit as X approaches. Negative infinity. Oh, that Lex. Since it's an odd polynomial, we know this is going to go to negative infinity. So no asking those least we now know in next we should find where the function is increasing such decreasing, so increasing slash decreasing So this says we need to look at the second derivative. They're the first word. So we get wide prime is going to be derivative with respect to X Oh, X Q. What excuse plus three X squared. Now we can go ahead and used parable to take the derivative of each of those. So be it. Three Next squared bluff six six x not sex 66 six x and then to the first power. Now, when this is strictly less than zero, it'll be Do you think I want it stripped? The grand jury will be increasing. So let's go ahead after this role so we can pull out three end of X, We get X plus one like this that you're here to ex exposed to and we know I just kind of looking at this. This is a quadratic. So it should look something like this. Insolent coefficients. Positive. Just using that. We can say that wide prime will be strictly greater than zero wet. We are on the interval. Negative infinity two negative too. Union zero to infinity. And then why prime will be strictly less than zero are decreasing on negative 2 to 0 the next thing they want us to find is our local max and open beds. So to do that, we want to set that equal zero so local max slash minimum. So we want to look at when why Prime is equal to zero and looking at what we have here, we can see that it will be possibly at X is equal to zero, and X is equal to negative too. And we can use this information here that we found about the increasing and decreasing intervals to see that we are increasing into. I hope so. My fellow out there so negative too so on the left of negative two were increasing into it and then decreasing. Actor. So this is going to be a local Max and X is equal to zero. What? Decreasing to the left of it and increasing to the right. So this would be a men and this will be a backs. So we have that the next thing will want to do is to find Kong cavity and points of infection. So I'll go ahead and have a little bit so khan cavity slash inflection once. So this means we need to look at the second derivative. So why double crime? Well, over here we found that wide Prime iss three ex swearing off sex sex. So taking the derivative of this function. So deed by the x of three X squared plus six x So they used power once again. So we would get six x to the first power plus and in derivative of X, it's just one. So we get six there and but we want to find changing con cavity. But we know that this here is just going to be a line that looks like that. So if we said this equal to 0666 um, we can just divide everything by sexual past and then subtract one over, we'll get negative one. Is it? So at this point here, who have a change in con happy And we also know just using this function here. So when why? Double prime is strictly greater than zero. Um, this will be calm, Cape up call on, and this occurs using the fact that we know the zero of this line is exceeded negative one, and it's increasing. That's it won't be from negative one to infinity And then why double promise like to be a messenger or con down on the other intervals. So negative infinity too negative. All right, So now that we have all this, it's just not one more time and start graphing everything. So you go ahead, Mark a little area right here. 1st 2 breaths. So go ahead and first plot all of our relevant information. So we know our ex intercepts were going to be at zero and negative debris, So zero and 123 negative three. And we also know that s R Y intercept is also at 00 So we don't have to do anything with those, um, then going to symmetry. While there was no symmetry asking Taub's there were no Assam toes, but we know are in behavior is going to be to infinity on the right and to negative infinity on the bus, uh, intervals of increasing and decreasing. We have that, but more importantly, we know we have a minimum at zero at a maximum, innit? Negative too. So this is going to pass through like this, come up until we hit some value for negative to come back down, and then it's going to touch this point because we have no other ex intercepts. So you have to bounce off that point. They come back up like that. So that uses five and six. Because we have our intervals of increasing and decreasing calm cavity at X equals negative one, which would be this point here. We can see that's going from con cave down on the left to con cave up on the right. And now the only thing we might want to do is to possibly find out what are local Maximum is right here. So to do that, we're just going to look at what, uh, of negative, too. Is so that's going to be negative eight plus well or four. So that point there would be negative to four. So this would be a nice little sketch of the graph using the method outlined in this chapter.

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