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(b) Find d*y for the curve dc~(t) =3 - [ v(t) = 92 36Hence determine where this curve is concave upwards....

Question

(b) Find d*y for the curve dc~(t) =3 - [ v(t) = 92 36Hence determine where this curve is concave upwards.

(b) Find d*y for the curve dc ~(t) =3 - [ v(t) = 92 36 Hence determine where this curve is concave upwards.



Answers

Find $ dy/dx $ and $ d^2y/dx^2 $. For which values of $ t $ is the curve concave upward?

$ x = t^3 + 1 $, $ \quad y = t^2 - t $

So in this question we have to find the square by by the X square For x equals to take you plastic and y equals two. The square Right. And in the second part we have to find whether the covers corn came up or corn came down at the equals to one right. So we can find the X by DT. That would be called a treaty square Last one and divide by duty that would be equal to duty. Therefore daylight by the X would be equal to. So they divided by three days where blessed one. Now we have to again differentiate this so they square right by this saturday square. We have to differentiate this with respect to the Mhm. So we will get it is if you buy the farm, so when you simplify we will get three T squared plus one. All the square. A new model will be 60 square plus two. Similarly, uh we can right day by the X. Yeah of day to day. Mhm. That is described by the X square. This one can be written yesterday by date a of divide by dx multiplied by data by the X. Right. Mhm. So when we substitute the values here we get 60 square plus two divided by three days where plus one is square and 85 G. Xs one a 10.80 square plus one. Mhm. Mhm. So when you simplify we will get the square by by the x squares -6 days square by two plus 2 divided by three day square plus one cubed. So this is the expression for this. Can I buy the excess square? Yeah. Now we have to find whether the car was can't give up or down. So we will find the value of this quote by by the X square At stake was to one. So I'm a substitute take was to one. We will get um -6 Plus two, divided by three plus one cube, which is approximately equal to zero point 0625 minus right. So since this is negative, so this is goodbye. By the X squared at stake was 2 1 his less than jail, which means that the girl is going to go down At the equals to one. Yeah right.

Yeah. Mhm. The problem is, find the Y the X and second derivative. Yeah, wide square over the X square for which values of T is a curve and curve upward X is equal to eat too. And why is equal to tee times two make two g. First we compute Divide the X. This is a cultural Why did he over the axe, did he for dy DT We use productive rule here. So this is one times e to negative t us key times it to negative t hams. Negative one. And the relative of wax is the to the T. Yeah. Uh, this is Echo two one. Minus t. Oh, uh, e two food here. Mhm. Now the second is a derivative. The wise Why our the X sky a definition? This is a cultural. He did. He Why, Jax? Oh, uh, the X The key. This is a call to for the numerator. We use caution rule. This is a co two e sure 40 ham. And and then this is 91 ums You two to t minus one minus t. Um, it took two t arms to and the X DT is e to the T simplify. This result they have this is is next to one minus to class. Fruity. Um, need to fruity. Uh huh. You, too. Wow. Mhm. This is a co two booty minus. Really? Um, we can consult into two t. Then the denominator is too rate. And, uh, lad, second derivative. You wanna square the axe? Go ahead. 3000 greater than zero. Then we have. He is greater than three over to one. He is greater than 3/2. The curve concave upward.

Hello. Welcome to this lesson. In this lesson, we'll find do I D x. Okay, So we find this by for the full finding the white tea, then again finding the x d t. Yeah. So d y t is given to us as one plus one of, uh, t That is called to t plus one all over T whoa, whoa. Okay. The next tender will find s d x d t. That is also given to us that that can be fine. Pass t that can be found as one minus one on t. So that is t minus one on t. Okay, So means that the white e X is given to us us. Do I d t s t blast one on t then dy dx DT s t minus one on t. So that is t last one. Go t plus one all over. T minus one. Okay, because this multiplies is just multiple eyes. This councils that Okay. Mhm. All right. Mhm. The next thing that we look for is a beeper that is this cleared y ds squared. This has given us d d t of do I d x all over the x d t. Okay, so here reward, differentiate this again with respect to t. Okay, so deed see? Oh, t last one, t minus one. That is sick. Or to G d t. So T last one quote, then t minus one. Okay. Oh, so that is given to us. Okay, What we do is that we will use the product through where we called one path. Then we differentiate the other path. So the first, but we differentiate this and we hold that. So this becomes one times T minus one. Okay, them Plus, we hold this, we differentiate the other part. Now we have negative one, then t minus one to the bar. Negative, too. So this becomes one over T minus one. Okay, Last T last one. Well, that's a negative. So minus the whole of this all over. T minus one squared. Okay. So we can let all of them be under one. Uh, so they will get t minus one, then minus the course in Both are naturally minus, so t minus one squared. Okay, so mhm. Oh, now this is where we have. Okay, four. Oh. Oh, Okay. So yeah. Negative. Two all over T minus one squared. All right. Yeah. Okay. The whole thing is for D d T. Yeah. Do I d x Now let's divide the whole thing by DX DT So you I mhm. Mhm. Yeah, Well, uh huh. But the ex d t would give negative to t minus one. Squared all over. T minus one. Yeah, t So this becomes negative. Two t all over. T minus one to the par three. Okay, so it means that described why the X squared is giving us Yeah. Okay, so for this to be for this to be greater than zero, then it means that negative two t should be later than zero. Okay. Oh, then it means dead. Mhm. He should be greater than zero. Wow. But at the same time for it not to be, uh, for it not to be on defined T shirt. Also less than one. Okay, so for it to be concave up, this tea should be greater than zero and should also be less than one. Okay, so this is the interval for the whole of the differential. The second differential. Who determines the conductivity of that? The diametric have should be the t the value of the T should be between zero and one. Okay, so this is the end of the lesson. Thanks your time. Oh.

The problem is, I find you wide the axe on dh second directive, you are square over the exploit or which Well, it was off t is a curve. Concave upward. Yeah, X is equal to squared plus one. Why it's going to be square plastique now it first away, compute you wind the X. That's if they go to the y. Did he over? Yeah, the y did. He is too t as long on that. The axe titties too. This is swallow off the wide. Yes. Now second derivative. You want where? Over your ex. Claire, this is a code two and he of Ugh. Why, Jax? Hola. Yeah, it's you. This is a call to Oh, the wind. The axe. Use caution through here, but this is two He square over on the numerator is who times. Who is he minus? Who did he trust? Juan? Times two over. The axe titty is too t simplify. It is the result we have. This is a true forty four t minus All times. T mine's too over. Oh, he's square. Twitty, this is E kowtow. Negative, Juan. Over Or hams. So three's power. We know one. The second narrative is greater than zero. Then it's a curve concave upward. So if he's why square over yaks return and zero. I mean, he is smaller on zero and the curve concave, not a word.


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