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0-/4 Points]DETAILSSCALC8 14.6.041.MY NOTESASK YOUR TEACHERPRACTICE ANOTHERFind equations the folloning- (z - 912 (a) the tangent plane(2,3, 11)(b} the normal line ...

Question

0-/4 Points]DETAILSSCALC8 14.6.041.MY NOTESASK YOUR TEACHERPRACTICE ANOTHERFind equations the folloning- (z - 912 (a) the tangent plane(2,3, 11)(b} the normal line (xlt) st) z(t))Need Help?

0-/4 Points] DETAILS SCALC8 14.6.041. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find equations the folloning- (z - 912 (a) the tangent plane (2,3, 11) (b} the normal line (xlt) st) z(t)) Need Help?



Answers

$39-44$ Find equations of (a) the tangent plane and (b) the normal
line to the given surface at the specified point.
$$y=x^{2}-z^{2}, \quad(4,7,3)$$

So for this problem, we are given that we have a equation of a surface here, which is a on this left hand side up here and then we want to find the tangent plane and normal line On this surface to the .335. And so the big thing to remember always is when we're trying to write the equation of a tangent plane, the equation of a tangent plane, we always need a normal vector. We always want the vector normal to our surface, the one that's perpendicular. And what's very helpful to know is that the gradient is always perpendicular to our um our our plane. And so when we construct a tangent plane at all, we always need that normal vector. And so the tangent plane, the normal vector to our tangent plane is going to be the gradient. And so the gradient F. Now, what does this gradient F actually mean? Well, the first thing we always need to do is make sure that our equation is equal to zero on one side. So for instance, we can move this over. It's not too much of an issue for this problem, minus three squared minus 10 is equal to zero. And this right here is our function F of x, Y and Z. And so if we want to compute our our um are tangent plane here, we just need to compute the gradients vector, which is f X, f, Y, f C. We take the dot product with x times x zero, y minus y zero, n, Z minus z zero. And we set this equal to zero because we know that these two things have to be equal to zero. This is our product here. They have to be perpendicular. So we can rewrite this pretty quickly if you wanted to. Um but the thing is we just need to find an fx fy and fz at the point X zero, Y zero is easier, which is 335. So we want to first compute fx computing fx right here is just going to be equal to four times x minus two. F sub Y is going to be equal to two times y minus one. N f sub Z of our function appear the river of expectancy to times z minus three. And all we're going to do now is plugging our points 3, 3, 5 to these. We get that FX is going to be equal to four times 3 -2, which is equal to four. F sub y is two times three minus one which is equal to four. Again, n f sub z here is going to be two times five minus three, which is also equal to four. And so we get here that our formula for a tangent plane is going to be equal to 4,44, which is our gradient dot product with x minus two as our x minus three y minus three, Z minus five. This is the point that we're worried about is equal to zero. So this could be a reason, this could be a reasonable answer to your question. But we can also just take the dot product here and write this a little bit more formally as for Times X -3 Plus four times y -3 Plus four times E -5 is equal to zero. And we write it want to write in our normal form here. We can distribute the four everywhere And see that we will get four X -12 Plus four, Y -12 plus four, Z minus 20 equals zero. And then we could see that if we write this as four X plus four, y plus four Z. This could be then equal to 12 plus 12 plus 20 which is equal to 44 lot of forces question. And so this is our answer. We might be able to even simple. I want even more by dividing out by at four which was the X plus y plus C is equal to 11. Whatever you think is more necessary. So this is our answer. This is our tangent plane, is the plane that sits right on our surface. And this is actually an ellipse oid. So this works really nicely. Now, the last thing we need to do is computer a normal line and our normal line here thankfully is the line that has the, that has our direction vector being the line itself, which is very handy. And so there's a formula we can use which is the symmetric lying formula over S Y and z minus Z zero over F sub Z. Which is the symmetric line formula here, which gives us a really nice way to express our our line here. And in order to do this we just needed how good fx fy and fz at this point which we already have which is 444. And the X0, is easier. Was our actual. So we can just write X -3 divided by four, Critical to Y -3, divided by four, Which is able to Z -5 divided by four. And this is our symmetric one. And that is our answer. And so this gives us our normal line to the to the plane which is the line that goes in the direction of the normal vector Through our .33, 5. And secondly, this is our tangent plane. And so the tangent plane always has a normal vector, that is which is the gradient. So, if you ever need a completely normal vector with a function, always think about the gradient and always make sure you move all of the function values and terms onto one side before you take the gradient, Like we did up here

Everyone to. Maybe I'm going to solve a problem. Number 45 here. A part of the question is, here's took x y that we kowtow X plus y plus that And the point is zero comma, zero comma one. So find the equation off times and plant at the given point so we can write like effects will be equal toe one minus visor. Arrest to exercise it effects at zero comma. Zero comma one. It's a call to what? F y equal toe one plus the decks. It has to exploit that. If Wyatt zero comma zero comma one will be one have that equal to one minus excellent, it is toe exercise it as that at zero comma zero comma one will be one. So find partial their way toe one in tow, X plus one and the wife plus one In those AG minus one. We will get like X plus five. Plus that if my insulin equal to zero, so now we need to solve be part of the question. So be part of the question is like we need to find a normal line equation, which is X divided by minus one, which is a call to why they were in by minus one, which is a call to exactly minus one divided by minus one, which is executed toe. Why you call toe that? Laying a smart That's enough question, thanks.

Today we're going to solve a problem. Number 41 here a party's functioning X Color by commas Article two Do we do Ex minister? The whole square Plus Why minus one. The whole square Plus that mine. Australia Whole square minus 10 Equal to zero. So for pain and plenty, question FX Is it going toe for X minus eight f x at three comma three comma five equal to four f Light is a call to for F y at three comma, three comma five If for exert a kowtow toes at minus six. Exerted. Three comma, three comma five years for fans and plan equation had specified point three comma three comma five years X minus three into four. Plus why minus three in the foot Plus that minus five into four equal to zero We will get like X plus y plus is at minus 11 equal to zero. So this is a solution off the past part. No, we need those order. Second part. The part of the equation is function if you call to two in the X minus two. The whole square plus why minus one. The whole square plus Zac minus trade the whole square minus 10 equal to zero. So effects a quarto for X minus eight FX at three. Com a tree come off five equal to four f y equal toe to like I minus two f y at three comma three comma five equal to four. Have that head X comma y co master, this does that minus six. Have that a three commentary comma five years four So normal in equation has fired at Trico Trico five years X minus three divided by four Which is a call to by minus three divided by four Which is a photo that minus phi They were in my foot. Thank you.

Everyone today we're going to solve a problem. Number 43 on a part here function F is a caldo x Y Z squared minus six. So F with respect to X equal to Why is that square F right equal to X that square? Have that equal toe to exercise it FX at the 0.3 comma to cover one, go to f y it three Command toe Come out one a goto three exerted three comma toe Come out one equal to 12 so times and plane is given by to window X minus three plus three Indo while minus to plus the well into that minus one equal to zero. We just two x plus three y plus towards that, a quarter 24. So this is a solution off first part. Now we need to find that the part of the question it's just normal immigration. So the part of the question is, we can right as solution is X minus trail divided by two, which is equal toe Y minus two. Divided by three, which is a call to observe minus one divided by 12. This is a normal Any question. Normal life. Thank you


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