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8. Let f(r) ={a I 4{-2,2} fe{-2,2} on thc interwl [-2,2: Then V(f) =...

Question

8. Let f(r) ={a I 4{-2,2} fe{-2,2} on thc interwl [-2,2: Then V(f) =

8. Let f(r) = {a I 4{-2,2} fe{-2,2} on thc interwl [-2,2: Then V(f) =



Answers

19-22 Find the component of $\mathbf{u}$ along $\mathbf{v}$.
$$\mathbf{u}=7 \mathbf{i}, \quad \mathbf{v}=8 \mathbf{i}+6 \mathbf{j}$$

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In this problem, we were asked to find the component of you along the so the formula for the component of you along t the dot product of you and fi over the magnitude of the So given that you equal seven i, which can be translated to seven comma zero and V is eight i plus six j, which can be translated to eight comma. Six. You put that in and you plug in the It's about him and we can simply solve. So the dot product of you envy is seven times eight, which is 56 plus zero times six, which is just zero. The dot product here is 56 that is in the numerator. I want to calculate the magnitude of e. We simply take the square root of eight squared. Add that to six squared that equals the square root of 64 plus 36 which is the square root of 100 equals 10. So the component of you along fee is 56 over 10 which is simply a 5.6

Either. So for this problem we need to verify the equation for the most probable speed and that equation is that most probable speed is equal to the square root of two times Bosman constant times that imperative for over the mass. Um So in order to do that, we need to start by able waiting the derivative of the number and is equal to that depends on the velocity with respect the velocity is equal to zero. And after this solving for the velocity. Now we know that this number Is in the following four. That is equal to a constant A times the speed is squared times the aspen M. Show of minus a constant, be times the speed squared. Now we need to take the derivative of this function with respect to the velocity. As you can see, we will have that that is since we have been here the product of two functions, we need to provide one first and live in the the other one constant. And then and plus uh everybody the other function living the other one constant. So in that case we are going to obtain that this is the derivative of the square with respect to the velocity times the exponential finance be the speed squared plus, living this now constant and hate derivative of the exponential minus b squared over. So if we do this, we will obtain that the real innovative of this function with respect to the speed is two times eight times the speed times the exponential of -7 square plus. Um Well, when you are going to provide the exponential function, we know that that is the same. The exponential function but seems we have in here some internal derivative, we will have that that is equal to -2 times a times b. And the internal derivative. Remember that in this case is two times speed. That multiplies dbs square. So we will attain B to the three. The exponential of minus B Be a square and this must be equal to zero. So if we set this equal to zero, we will have that. This is equal to two. And the exponential councils, we can cancel the espen m shows because it is present in both sides. So we will have the following expression Um this one right here. Well, this is 2, 3. As you can see, we can cancel the AIDS in here. And from this we solve that The speed squared is equal to one over B. And we know that the constant be is equal to the mass. Overt two times Bosman constant times that Imperatori. And since we have in here that this is and the inverse of this, we will have that. Then the inverse of B is equal to two times both men, constant times the temperature over the mass. So we will found that the most probable speed is the square root of two times the balls man comes sometimes the temperature over the mass. And as you can see, we have shown the relationship, we wanted to show. So this is it for this problem.

And this question were given vectors U and V, both of them in terms of I and J. We want to find the component of you along V. It's a component of you Along V is given by you dot v over the magnitude of V. Now these are written in terms of I and J. But if you would like to, you can change the format. If that's easier for you to look at, this is the same as seven Negative 24 and over here would be zero because we're missing that I and then just one. So let's find the dot product first. You dot V is when we multiply corresponding components together. That would be seven times zero, because those are both the first components. Plus, the 2nd 2 multiplied together so negative 24 times. One said it zero minus 24. So that's negative. 24. So I can go ahead and plug that in for you dot Be on the top now to find the magnitude of V, that is the square root of the first component squared, plus the second component squared so as the square root of zero plus one and the square root of one is just one. So plug that in there for the magnitude of V and negative 24 over one is just negative. 24. So that is your answer.


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