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Given f(a) = 7r + 9 and g(w) = Vx S 4, find f(g - 1(2))...

Question

Given f(a) = 7r + 9 and g(w) = Vx S 4, find f(g - 1(2))

Given f(a) = 7r + 9 and g(w) = Vx S 4, find f(g - 1(2))



Answers

Let $\mathbf{v}(s)=s^{2} \mathbf{i}+2 s \mathbf{j}+9 s^{-2} \mathbf{k}$ . Evaluate $\frac{d}{d s} \mathbf{v}(g(s))$ at $s=4,$ assuming that $g(4)=3$ and $g^{\prime}(4)=-9$.

In this problem of line integral. We have to calculate the line integral of the vector field along the line between the given points. And we have, given the force field we have is equals two. Fs equals two six I-. This is six. I-. So when do And we are moving from 002 76. So this is 00 276. So let's draw these conditions. So let's suppose this is the origin. Say this is origin and this is y axis and this is X exit. We are going from zero point oh and we have origin 00 to a point which have X is equal to seven and y is equal to six. So here from 00 to this is the point. We can name it as 76. This is point B And we have the coordinates 7, 6. So this would be a straight line joining this. And now with the field the slope of vector field is so putting the y coordinate. So here slope would be put in the white coordinators in minus seven, divide with six. And slope of this line. Or we can say the direction would get from A to B. And slope would be here say seven divided with six. And where is the angle between these two slopes. So this was everyone is equal to minus seven, divide with six. And second slope M two is equal to seven divided with six. Sorry, this really would be 61 with seven. This way we will be six divide with seven. And here we would have six anywhere with seven. Now when we multiply, so we have anyone multiplied with them too. So -7 divide with six, multiplied with 61 with seven, which gives you -1. And we know that if two slopes when multiplied of a straight line, Two slopes are of a straight line. And if multiplied gifts product is -1. That means both are at perpendicular. Or we can say both are at 90°. So that's why the slope will be like this. All the time direction will be say perpendicular. So this would be All time perpendicular. So that's why if the angle is 93 that wins, tita is equals to 90 day. We and all the time dot product would be equals to zero. So we have the value of line integral is equal to zero.

In discussion. Okay. Says that the given vectors are orthogonal so you get to it. Air time, first ball game and the referees. Why minus you do now The formula of dot product of two vectors is supposed directories. Hey, when I blessed a blue jay and B that there is you and I plus be today so their door product is a better don't be that there is even B one plus two. We do now it is given that the two vectors are orthogonal, so there don't product will be zero Now we'll find the door product, you better go be that equals two 80 plus four j So y minus K. J Now we will apply them to the formula eight multiplied by two plus four multiplied by n minus K consulting this figure 16 minus four K. Now the two vectors are also going. And so the door product is You wanted to be back very close to zero 16 minus four k close to zero. From here we will get the value of K, which is okay. It was the extent close to four. Thank you

In problem three. We want to compute the normal view the normal vector V. And the dot product of the victors you envy? Where are you? Equals two. Oy. Palestinian J minus 12 K and V equals three. Oy plus four. Key. Let's start by the normal view. We can get the norm by taking the square root of the submission of each component square. It's two squared 4. 10 squared 100 Plus 12 sq 144. Then it equals To multiply by square root of 62. Let's get the normal v square root of each component square. It's three square nine plus zero square zero plus four square 16. Then square root of 25 gives a five. Finally U dot V equals the submission of the multiplication of each component. Then we have two plus two, multiplied by three of six. Then multiply it by zero. Give zero -12, multiplied by four minus 40 it. Then the answer Is -42.

If factors U and V are orthogonal you that we is zero. So we will find you that be first. Then we started to be zero and soft. Four k first, the one you that B is one times two plus K times negative five minus three times four So to minus five K minus 12. This is negative five K minus 10 because U and V are stock. No. So this is zero. So for okay, minus five k is 10. So Okay is negative two Next one you that be is two times six plus three k times negative one plus negative four times three and one time seven on day, five times two k So 12 minus three K minus 12 plus seven plus 10-K you go to seven K plus seven again, this is zero. Seven k is negative seven. So K is negative one


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