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Use the guidelines of this section to sketch the curve.
$ y = \sqrt{x^2 + x - 1} $
see solution
Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 5
Summary of Curve Sketching
Derivatives
Differentiation
Volume
Campbell University
Harvey Mudd College
Baylor University
University of Nottingham
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we want to sketch the curve. Why is it to X squared plus X minus one square rooted. So in the structure we have this laundry list of steps we should follow and probably want to sketch the rap of something. So the first step being, we need to find our domain. Now we know the domain of something inside of the square. Root needs to be restricted such that the inside is going to be greater than or equal to zero. So to do that, we're going to go ahead and factor the inside. So, unfortunately for us, this doesn't factor very prettily. But, uh, if you do use like the quadratic formula, you can they grow fruits of this and see that this factors to be negative one or X minus negative one minus square root of five all over two times. Put another big bracket X minus negative one plus the square root If I over too all square rooted. And so if we go ahead and set the inside of that greater than equal zero, we would end up getting that Our domain should be negative. Infinity two negative one plus square root up. Five over too now, square brackets, because we can plug this value into that union with the other one, which is going to be one minus. That's where we're five over too to then vanity like this year. Actually, those be switched up here. Those signs so should be plus and minus. All right, so these here should be our to the art of it. Right now, the next thing we're going to do is go ahead and find our intercepts so our intercepts will be so let's go ahead. And first notice that our why intercept isn't going to be defined since it is not in our domain. So we do not need to find our wider set, but at least our ex intercepts. So X intercepts are going to be where why is equal to zero. And that's just going to be our two values that make this doctor. So that tells us X is equal to negative one plus the square root of five. But but you, as well as negative one plus the square root of five or two. So those will be our Q X intercepts. Now let's go ahead and see if this function hasn't symmetry well, normally we really wouldn't think the square root of something or electronics are really symmetric or periodic. I should say so. Let's just go ahead and check to see if dysfunction is even or odd. So effort negative X is going to be so we'll be negative. X squared minus X minus one square Reuben. So it's gonna be X squared minus X minus one. And this here is not equal to FX or negative books. So we at least know there's gonna be no symmetry, at least non about E. Why access for the origin? All right now, the next thing we're going to want to do is to look for us in tubes. Whoa! In this case, we know there's not really going to be any acid trips, but we can find our in behavior. So let's do that. So the limit as X approaches infinity of F of X is going to be so looking at the original function we have on the inside. Essentially, it's just going to be X squared, so X squared goes to infinity. Since X squared dominates X and negative one eventually and then square root of something, or to infinity just the infinity and using that same logic on the inside, I'll just be X squared. So as excuse the negative infinity that would go to positivity and scrub room that would still go to. So we can actually say that the in behavior for this ward plus surmise and any should go to infinity. All right, The next thing we want to find are our intervals where the function is increasing and decreasing, as well as any local Max or men's. We may have so to do this, we need to figure out what wide prime is going to be equal to. So it's going to do that on the page. So I have Why isn't too X squared sex? Nice one. And now when I take the derivative of this, I'm going to need to use General. So the derivative of outside remember, the Square root is really just 1/2 power. So this is now going to be 1/2 X squared plus X minus one to me negative 1/2 power and then we need to take the drove over inside function, which is going to be two X plus one so we can go ahead and rewrite this now as two x plus one over two X squared prospects minus one minus one square root. So the final critical values we could go ahead and set this function equal to zero and wouldn't be solved. For that, we would get two X plus want is equal to zero. Since remember, nothing plugged into the nominator will ever make a function equals zero. And that tells the specs busy, too negative one. How? Oh, right, No, Something else we would also want to check is where our function is undefined. So it is undefined when the nominators equals zero and those were X is equal to. So I'll just say negative one plus or minus. That's where we have five over. But since he's our end points, we don't really need to worry about these being are critical values. So since I can't really be a and about you, it can't be a max or event, since they are the inputs. So we don't need to worry about that. But do you remember In general we do need to check those low. So we had why Prime is eager to two x plus one over to square brute. Oh, X squared plus X minus one. All right, so let's go ahead now and actually find where this function is strictly logical. Zero. And this is going to be Why so why prime is strictly larger than zero, or the function is increasing up, not up increasing on. So I just went ahead and solved this beforehand. So this should be the square root of five minus one over two two. And we know why. Prime will be strictly less than zero. Tells us it will be decreasing, or this would be negative. Infinity two negative one plus the square root of five over too. So before we found that Exeter negative 1/2 is a possible, uh, maximum or minimum of this. But if we were to look at what this value one plus word by liberty is this is not and don't make So because of that, we actually won't have any maxes Airmen's at all. And that was probably something you should have checked in that first step to see if this was in our building. All right, we have that now, and the last thing they tell us to do is to look for Con Cavity and any inflection points we may have. So that means we want to find what why double Prime is equal to. So let's go ahead and find our second derivative. So why Double Prime is equal to so to take the derivative that we're going to need to use change will, I mean a questionable as well as trainable. So remember, Changeable says so. First, I'm not back to that 1/2 out front just so I don't have toe carry it around everywhere. So question rule is going to be so low. X squared plus X minus, one times the derivative of what's in our new minor two x plus one minus into the opposite order to express one times the derivative of X squared plus X minus one square all uber what we have in the denominator squared so x squared plus X minus one, and then when we square that you should actually have absolute value in the denominator, since we know the square roots always because it and then swearing that. But on our domain, this value will always be bigger than equal zero. So we don't have to have to worry about that the absolute values and we can just ignore them in this case what? And if we get through all of this algebra here, we should actually end up with actually, let me take the drip. That's first again having myself. So the derivative of two X Plus one has been to X, and we already found out that the derivative of the square root of Expert plus that's nice one is going to be two x plus one over too X squared plus X minus one square root. And since that's just the first riveting now if we go ahead and simplify all this algebra here, then we're gonna be left with negative five over or finds X squared plus X minus one to be Billy House power. So if we were just at this function equal to zero, well, it's not going to be equal zero since end the new mayor. We only have negative five. And just like before, the only place this is undefined is when our denominator is our end point. So we don't need to worry about points. So for this we would have no points of election, and even further we know that this here should be strictly less than zero. So our function will only be calm cape down. All right, So go ahead and write all bad movie, you know. So our second derivative was negative by over or times X squared plus X minus one to be three house power. And just like what you said before, why double prime will be strictly lessons there or con cake down all of our entire domain. Excluding the end points. So negative infinity to negative one cross the square root of five. But you union negative one plus the square root of five over too. To then vanity and just say appear wide, double prime Strictly greater than zero. Never. So there's no intervals for that. All right, so now that we plotted all this information, we could go ahead, actually start sketching our ground. So to start off, we have our two intercepts here and here, this one being negative one plus. Describe the five and the other one negative one plus the square root of five over, too. We're going to have no local Max's airmen's. We know he should go to infinity on each side, and we know the function used to be con cave down on the entire. So we know so to the left of our graph, it should be decreasing. So it should look something like this and just going off to infinity like that, keeping where it's conking down and on the other side is going to be increasing and it still needs to be conking down. So look like this. They're so we could maybe go in and put one or two points on the decided to maybe make it look a little bit better. But we've already grabbed all the relevant information, so I would say we could just go ahead and stop here for our sketch.
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