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2-In(xl+y2) , xerstsin(t) , y=reJrst then the value of @z dt at (r,s,t)=(1,-1,0) is...

Question

2-In(xl+y2) , xerstsin(t) , y=reJrst then the value of @z dt at (r,s,t)=(1,-1,0) is

2-In(xl+y2) , xerstsin(t) , y=reJrst then the value of @z dt at (r,s,t)=(1,-1,0) is



Answers

If $ x^2 + y^2 + z^2 = 9, dx/dt = 5, $ and $ dy/dt = 4, $ find $ dz/dt $ when $ (x, y, z) = (2, 2, 1). $

A little radio. What Back We're on question 2 28 Okay, in a tapestry. Chapter five, Given that C is a function of X and Y X credible, but ex intern is a function of tea and that's gonna be t squared. And why and why? It's also fun of tea. It's take yours. What is what ziti, Right? So thes e t. It's gonna be the first personal Z with inspector acts times the derivative of X suspected plus for special of +06 Why times the derivative of Wadi It's 50. So what is the first partial of zero respect, Maxwell? Why is a constant The derivative of X squared is for Texas to X and tons of whites is to listen to its way was a driven of exit t Well, through t squared is to t all right, Plus, what's the derivative? What's the first partial busy inspector? Why? Well, that's like two. Why the rivets with why would be too so no was he squared is a constant menace. Left what? A zero. And then the drew was sees three square. Okay, and that's an answer. But why don't we write it? Four in terms of tape four times X, which is t square times. Why? Which is take you to the fifth time sees 26 was three times x squared. But that's gonna be t squared T squares for to see the fourth Dynasty Square To see the sixth you get 76 children too, Nagel. I hope the hope that was helpful to make a comeback when every stock bye bye.

Okay, So for this problem were given the equation Z equals E to the one minus X y Power were given that X equals t to the one third power and that why equals t to the third power. Okay. And so what we're asked to find and I'm gonna do this in a different color. Is DZ over DT And since the equation is he doesn't have a t variable, we're going to use the chain role so dizzy over DX d x over d t plus d Z over d Y de y over DT. Okay, so now let's go back up to the top and actually go ahead and find thes values. All we have to do is plug them in so d Z over d x. So I'm gonna have e to the one minus X y power. Then if we take the derivative of one, that's zero so we could ignore it, and then the derivative of X Y or negative X Y in terms of X is gonna be negative y And then I'm gonna find d Z over d y. So if I do the same thing, so it's each of the one minus X y power. This one's going to be negative. X and then D uh d X over D t is going to be one third t to the negative two thirds and d y over DT is going to be three t squared. So then all we have to do is plug these in someone I have negative Y e to the one minus x y power times, um, one third t to the negative two thirds power plus negative x e to the one minus x y power. And then we have three t squared already. So what I want to do is I want to go ahead and plug in, um, some values. So what we're given, um, up at the very top, actually, me scroll back up. So we're given that, um, excess t to the one third. Why is t to the third? So what I could dio is I can plug these back in, so I'm gonna have negative wise. So negative t to the third e to the one minus, um t to the third and then t to the one third and then one third t to the negative two thirds and the negative t to the one third e to the one minus t to the third t to the one third and then three t squared already. So what I want to dio is I wanna add so I'll have negative one third and then I have t to the third and t to the negative two thirds. If I were to combine these, I will get team I broke. Gotta add the exponents when I get t to the seven thirds and then e to the one minus t to the third, um, times t to the one third, which would be three plus one third would give me t to the 13th plus negative three. And if I add t to the one third plus T or times T squared, which would add the exponents, I get t to the seven thirds and then e to the one minus t to the 13th because that's gonna be the same thing as the other one. So now I'm starting to see some similarities. So again, if I add negative one third and three, I'm gonna get negative 13th t to the seven thirds and then e to the one minus t to the 13th, and that would be my final answer

To find the value. We can use the inverse normal function because we want to obtain euro value. Given a probability, the inverse normal function utilizes the area under the graph from negative infinity to a Z value. So we want to make sure to find the area to the left off the Z value there were interested in. Given an area off 0.9671 we can use inverse norm on a graphic calculator and obtain the answer, which is 1.84

To find a Z value. Let's extract somethin's from the question first. Since the normal distribution is symmetric, the area to the left off the mean zero is the same as the area to the right off the mean. Also, the total area under the curve is one, so the area to the left off zero is 0.5. Next, we want to use the inverse normal function eventually because we want to obtain the raw value. Given a probability, since the inverse normal function utilizes the area under the graph from negative infinity to a Z value, we want to represent the shaded area under the graph. In the same way this means that we should have the shaded area will need to the left off the devalue that we want to find. We can first finally area to the left off Z, which is 0.5 minus 0.4175 as given in the question and that gives us 0.8 to 5, is that we can use inverse norm on a graphic calculator and obtain the answer, which is negative. 1.39


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