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Find dyldx and dkyldx?_ x = 2sin(t) , Y = cos(t) , 0 < [ < 25dx dFor which values of t is the curve concave upward? (Enter your answer using interval notation...

Question

Find dyldx and dkyldx?_ x = 2sin(t) , Y = cos(t) , 0 < [ < 25dx dFor which values of t is the curve concave upward? (Enter your answer using interval notation:)

Find dyldx and dkyldx?_ x = 2sin(t) , Y = cos(t) , 0 < [ < 25 dx d For which values of t is the curve concave upward? (Enter your answer using interval notation:)



Answers

Find the interval on which the curve $y=\int_{0}^{x} \frac{1}{1+t+t^{2}} d t$ is concave upward.

The problem Use. Find that the wide the axe on dh second directive a tive by square over the X square which, while some thirty Is that sure Concave upward. You're actually physical to Santy. Why is he goto sign two champs, Tio, He's between the right time. First the computer T y the axe. This is the coat with you. Why he he over? Yeah, thanks. Yeah. Why did he is assigned hootchy times two on the X ditty is negative. Fine. Now we computed the second derivative. This's too Yeah, you wide yaks over Jax to keep Oh, so the DDT divide yaks? We use caution rule since he called to negative sign Squire on it is year. This is two times make you sign to teach ums too. I'm still negative. Fine. On the minus two hams. Sign to tea. I'm selective. Sign she over the ex titties. Negative sign He and this's iko too. Rosa numerator. This's or sign he sign. Who is he? Plus, who'd hamzik assigned? Key Sign to a T over negative sign. Hey, two, three pounds Now for which values off T is a curve. Concave upward. We now want the second narrative is security and zero Then a curve on cave upward of a light He y square, x squared riches and cereal And then now this in clarity we know one cheese between zero and pi. Scientist E is always riches and zero, isn't it? So the denominator is always s'mores and zero things. That a lot. This one is great, isn't zero So it is equivalent to see numerator is also smaller than zero. So we have all time. Sign he I'm through with you. Us two ham signed signed here. Sign two fifty is smaller than zero. Now we use shoe tree God in peace assigned to Teo Is he going to two times? Signed signs on DH Sign to Teo Record sign. He's squire minus sign Key square. This is total to Tom's call sign. He's square minus one by this two identities. We have four time signed here. I'm too damn sign Sign Last two times. Assigned here times two times. Call sign. He swam once one smaller than zero. We help ate ham. Signed his squire with hand use one minus co sign. It's quiet in times Signed you last two times. Call Sign He times two times consigned. His squire, minus one is smaller. Zero Was this one the kind, right? This's two times call sign he Terms or Tom's one Minus was fine. Esquire class two times consigned. He's quiet, minus one smaller than zero. This is two times call sign here three minus two Assigned. It's Quinn, Smaller zero. Since we know Sandy Square and smaller than one So two times cause anti Square is smaller than to this part is greater than zero. So we have we'LL sign he squire Sam Qi smaller than zero. And since tea is between zero two ply So we have was from this one we have He is the last to know Hi. Over too. Too high now one t It's between pie over to pie The curve. I'm careful upward

In this question, we want to find the intervals of T where the graph is concave upwards and when the graph is concave downwards. So before we do that we need to know a few things. We need to know that for compatibility, we need to look at the second order differentiation. But first we need to know the first order, which is the I D. X. And there is Dy over G. T divided by the X. Over PCI. For the second order, it will be D over the T. Of the dy dx divided by dx DT. Now, if we are looking at concave upwards mhm it will be like this. It means it's a minimum points. So our Second order deprivation is Square and zero. So when we set this condition will find the in the world T. That was satisfied. This. When his concave down what's in this we're looking at maximum point, our second order differentiation is actually negative. So I will find the interval of T where this condition happens. So let's find our dX DT and Dy DT first. So dX DT Okay, Since four is multiplied to the co sign for can be set aside of course 90. When you're differentiate with respect to you, I'll just get minus 70. Mhm. So it would just be minus fel sai inti for Dy DT since two is multiplied to the sign, it can be kept aside Now 70 when a differentiate respected T or just get co sign T. So our dy dx Yes. Mhm. Dy DT which is to call 70 Divided by the STT, which is -470. So this gives us -2. Put engine key. Let's look at our second order. Okay, Our second order says that we have two different shape. We respect to t the dy dx and there is minus half contingent E divided by the dx DT over here, which is this minus 4 70. Okay. All right now differentiating minus half contingent T the minus half. Okay look the minus can be canceled with this and the half can be brought up. So let me bring out the half. Now I'm only differentiating contingent E. That gives me minus who's second square T over 4 70. No um Half and the four here they would interact and I will get 1/8 With the minus sign of -1/8 here. Now 1/70 will become Corsican so cozy conjoined because he can square here I'll get Corsican Cube t. Now at this point it is a lot easier. We were to just grab the second order differentiation. Now if you were to grab it you will realize it looks like this. Okay. All right. Yeah, there is a simple pie here And then another isn't to two pi and this is like this and like this. Okay, this is great finger. Second order differentiation. Yeah. So this is the this one is the 2nd Order Differentiation Graph. Now we're only clearly looking at the weather is possible negative which region. So for concave we are looking at the second or the differentiation being positive. So this is the path they were looking at. So our Now this is tear not excellent. No, because it's in terms of T. Were graphing in terms of T. So our T. Is between pie and two pi for concave down. We'll complete the words. We're looking at the point where our second order differentiation graph is negative and this is the point here, right? This is the negative. So our t. is between zero and high and we're done.

Okay, so we're given the function g of X here and were asked to find if or where this is Khan gave up or con cave down. So to find that these intervals of con cavity, we're gonna take the second derivative. But before I do the second derivative, we have to do the 1st 1 Now it's to take the derivative of this. It looks like we want to use the fundamental theme of calculus. The problem is that we need a constant and the lower bound and a very real in the upper bound. So to fix that, we're going to flip the bound. So have zero in the lower bound and X in the upper bound. And then we make the whole integral negative. I'm actually I'm gonna put that negative on the inside, too. Negative t over two squared plus one DT. So now the derivative just follows directly from the fundamental theme of calculus and is going to be negative. X over X squared plus one. So now we need to find the second derivative. And to do this, we're going to use the quotient rule. So we dio the derivative of the numerator first which is negative one. We leave the denominator alone minus we leave the numerator alone and we do the derivative of the denominator, which is two X all over the denominator squared. So expert plus one all skirt. Now, if you simplify this, we get negative X squared minus one and then plus two x squared all over X squared, plus one screen on. And then this simplifies to just excrement minus one over X squared plus one scraping. So to find the intervals of con Caveney we're going to find where the second derivative is zero. In this case, the second derivative of G equaling zero implies that we only have to look at the numerator equaling zero. So expert minus one equals zero, which we can solve this to see that that, uh, at plus or minus one. And now we're just going to make a number line and so we can see negative one and one. We're going to test all these intervals. So if I test simply test when X is equal to negative two. So if you plug g of native to into the second derivative, what we're going to get is a positive value here. Okay. In this case, um, we don't actually need the value, which need to know that it is positive if I picked something in between. Negative on in one such a zero. So you will test X equals zero okay than the second derivative at zero is going to be a negative number. And then last, we will do a test. X equals two. And when I take the second river, they've had to We will get a positive number. So what does that tell us? Because we're dealing with the second derivative here. This tells us that it is con cave up that g itself. The original Gs Khan gave up on the interval. Negative infinity to negative one. And so he's a union. The interval from one to infinity. And then it is con cave down on the interval. Negative 1 to 1.

We know that why is crank it down on the specific intervals where why prime are the derivative is decreasing, So let's calculate the derivative first. Now we need to figure out where y double prime. In other words, the second derivative is less than zero to figure out where wide prime the first derivative is decreasing number off one G myself, you won over G squared. You can use the quotient rule because we have a numerator and denominator over here to figure this out. Zero equal to X squared plus four ax. We get access. Negative floor comma zero X squared plus works is upward opening problem. The word sister problem that opens upward there for the integral neck. The interval negative four comma zero is negative. So, in other words, negative for commas. Here is where it's con caved ounce. That's our solution


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