All right everyone, welcome to our lecture on tangent planes. So before we talk about tangent planes, let's remind ourselves what we remember about tangent lines. We had our graph in two dimensions some function of a single variable, and we drew our generic picture like this. We picked a specific point, let's call it a, and then we said, let's approximate the function at the curve with a tangent line. And at that point, we did the best we could to draw a tangent. And that line was great, we used that line to approximate the values of the function, and it was really close to the line. And when you draw it, you see there's an overlap, and that was what we did before. So the question is, what happens if we take this idea and expand it into multivariable functions? So let's draw a 3D graph. So I'll always draw the x, y, and z axis. And our generic 3D graph, of course, looks like the magic carpet sort of floating out in space. And this is the graph of two variables f(x, y). Give me two numbers in, I'll give you some some output z value. And we're going to do the same idea. We're going to pick a point in the domain. So maybe we'll pick a point, but now point of course isn't a, it's a given x, given y, we'll denote those x0 and y0. You can imagine that if you throw it into the function, you get some specific output at the. Now, the tricky thing is that at this point, there isn't just one tangent line. You can imagine if you were to draw, you could draw a tangent line that comes sort of north-south, and you can draw one that comes east-west. And you could even rotate these lines to get infinitely many. So at any nice point there are infinitely many tangent lines. However, if you start to think bigger and think well, what do I get if I put all these things together? You get not just a tangent line, but you get instead a tangent plane. And that tangent plane touches this magic carpet, this bendy graph at one point, just like the tangent line touched our graph at one point. And so we have, instead of the tangent line, the tangent plane. We have a couple goals with this tangent plane. The first, of course, is to find the equation of this tangent plane. And then we want to sort of use this tangent plane to approximate the function as well. And so let me give you the equation of the tangent plane. To get the equation of the tangent plane, we're going to give you some multivariable function z = f(x, y). And we're going to give you a given point P, we could denote it as (x0 and y0). So it's like give me the function of the tangent plane to some f(x, y), x squared plus y squared, whatever it's going to be, at some point, (2, 4), (1, 0), whatever, whatever, whatever. These are my inputs, this is what's coming in. Here comes the formula, it's going to be an equation, so there's going to be an equal sign in it. So it looks like this, it looks like z-z0 = f of x(at the point P). So this is the partial derivative of the function f(P) (x-x0) + f of y(P) (y-y0). Stare this for a minute, let it sink in again. Again, this is an equation, so I absolutely should have an equal sign. A common thing that people do is they forget the equal sign or they said it equal to 0 for no good reason. But it's z-z0, so z0 is not, I guess, exactly given, but remember this is the output when I plug in my x0 and my y0. So I have the function, I have my points, and I can plug in, this number were going to go ahead and find. This is a multi-step problem. Will do examples with these with real numbers in one minute. Here's f(x, y), this is the partial derivative, partial derivative of the function with respect to P. Here is the partial derivative of y with respect to P. And one thing I just have to call out, plug in the value. This is some number, this is some number, these are coefficients, we're going to plug in, plug in, plug in. Okay, let's do an example. So keep this formula handy, we'll write it again the next slide, but let's do an example with all these things, ready? So let's find the equation 4 cosine, (pi over 3, 1, 2). All right, let's go through these steps where we're going to find the equation of the tangent plane. Eventually we're going to use the tangent plane, and this will become one step of many, the first step of many. But right now we just have to find the equation of this tangent plane. Our function f(x, y), this is the multivariable function, it is f(x, y). Our point, (x0, y0, z0) is given. So in our formula to start seeing things as they appear. They may not be explicitly labeled, but it's our job to sort of put this all together. As we go through this, I kind of want to approach this as a recipe, what's our first step? We're always going to find z0. In this particular case they were kind enough to give it to us, it's this equal to 2. They don't have to give it to us, they could have just given us x and y, and we would have had to gone in and plug in to cosine x and y. Plug in at pi over 3, times by 4, and then get back 2 anyway. But at least write it as a step so you know it's gotta be done. We are then going to find the partial with respect to x, and since Marshalls let's go ahead and get the partial with respect to y. If you want to pause the video here for a second, practice your partial derivatives, that's fine, but we can do that if we're going to do this partial with respect, we have four comes along for the ride. Derivative of cosine is- sine. We keep the inside of the same and then we by the chain rule, take the derivative with respect to x. So xy pops out, I'll write it out here and partial derivative with respect to y is very similar. We get a -4 sine xy and now an x. So we want to find these partial derivatives. Now this is extremely important. This is a step that everybody forgets. I'm going to put in red, so we call it out. Now we plug in. We plug in were going to get f partial with respect to y plugged into my point P. The point P that's given here. So this is really given point, and we're going to find this, and we're going to find the partial with respect to y at P. So we plug in at the end of the day these should be on number. They should be a number. When I plug in, you can check the math on this. You get -2 route 3 its good or common trig values and I plug in at power 3 and 1 and you get -2 route 3 over 3 times pi. So remember what this is? I'm just taking -4 sine of xy times y and I'm plugging in at my x not y not value my pi over 3,1 and I do the same thing for the partial with respect to y. Last but not least, we use the formula. Then we can write it out in full generality, or we can go ahead and just plug it in. It was on the last slide. Hopefully you have it so in the interest of time, let's just plug it in. I'll say it so it z- z not so good old z- 2 equals, then the partial derivative of x plugin. So that's -2 route 3. Parentheses x- x not and then plus the partial derivative of y plugged in. So that's going to be- 2 root 3 over 3 times pi, and then times parentheses, y minus y, not which not the world's most prettiest numbers in all honesty, and I guess we could clean it up, but there's really no need to. The point of this is sort of go through this four step process, put it all together at the end of the day we come back with a formula for the equation. Now I said before after we get the equation of the plane we want to use this equation of pane. So now we want to use this thing to approximate the function. We have difficult function we want to do linear approximation. So before remember we had before draw the picture one more time, but I'll focus more on the line now. So we have our single variable function f of x. We pick a point, call it a white, not and we have our tangent line. Red the curve take the equation of this line since starts with an l, will call it l of x. And remember how we approximated this? Remember the slope of this line is the derivative, so the slope of this line is the derivative f prime and a. And I also had a point remember point that we had the x values a the y value was half of a. So putting this altogether I can get the equation of the line linear approximation was given by the point slope formula. So friendly reminder of the point slope formula y- y 0 equals mx minus x0. If I start to put the equation of the line using the point slope formula with our given terms. Here's a sloping the derivative and our point is f of a. We get something that let's us solve for y, the y value. The function is given by l of x, so you can move things over so particular its move the y value over. So we have why 0 + m x- x not. Just look at the equation, the point slope and think of what your equation of your line is your y value. Plugging this all in for our values. Here we get it, why 0 the given y value is now f of a the slope m is the derivative at a and x- 8. This is our linear approximation. This is the for club values that are close to this linear approximation. Let's do approximates the actual function and we use this. We had nasty functions or like either the two or something like that, or natural log of like. How does your calculator actually work? What are ways that we can get decimal approximations not taking the calculator for granted? So this was what linear approximation was in one variable. Now we have the same idea, so we're going to draw the magic carpet floating in space. There it is we're going to pick a point. We're going to draw the tangent plane at that point. And now instead of a single variable tangent line, we have a multi variable tangent plane. We'll call it L of xy. This is a linear plane. It's a flat bunch of lines together, and for all the same reasons we have the equation of linear approximation. It's going to follow exactly the same in the equation for the tangent plane, just solving for z. And so z is our output and we get that the linear approximation, the equation of the tangent plane is z naught plus f of x, plugged in at your point p. You're given a point. X minus x naught. Plus half of y. Plug in at your point p. Y minus y naught. This is the equation of the tangent plane, and more importantly, this approximates the function. It's super important that it's only a good approximation if you're picking values around p. If you're picking points around p. You could even see you in the single variable case, the further away you get from the point that you're studying, the nice point, then the error tends to get large and this approximation tends to get not too useful, so keep this formula, this new formula in mind. Compare it though to the equation of a tangent plane. Hopefully you realize it is exactly the same. All I did was solve for z. So if you know one formula, you know them both. Now let's go ahead and use this thing. So here's what I want you to approximate. Nasty here, 1.1 in parentheses times the natural log. Of four times .05, plus 1.1. Close brackets squared. Now I know what you're saying. Well, can I just plug this in a calculator and get a decimal? Yeah, but you're kind of missing the point, you're like, what's the calculator doing behind the scenes? And let's just go through and see an application of this. Okay, so we can compare it with the actual value of the calculator after, but the way I want you to think about this function is this is really a, there's two points that are showing up here, right? There's 1.1, that shows up twice, and then there's .05. If we replace these values with an x and a y, we can think of this as a function f of xy. As x times the natural log of four times y plus x. And we could put that whole quantity squared. And we can use rules of logarithms to even make life a little easier, and we can totally bring this out front if we want, just take advantage of some rules of logarithms, and we have 4y plus x. It does not matter which number you call x and which number you call y, and obviously this thing has to be cooked up nicely so that we have two points. And if you look at 1.1 and 0.05, you can say, well, these are kind of nasty points. Let's, I don't know what the graph looks like, but we're going to try to study a nicer point as we're trying to think of a picture of what we're doing. We have this kind of not so nice point. If we pick a nice point p that's close to 1.1 and .05 and the points 1 comma zero comes in mind here. It's just nice points. We'll study p, will look at the graph above p, we'll get the tangent plane above p. These are all nice points. I like plugging in zero. I like plugging in one. And then we'll use the tangent plane to approximate the not so nice point. Okay so step one is to think about it. So sometimes they give you the function immediately, sometimes they don't, but one is, I'll just, so we can start on our steps here, is get the nice point. Get the nice, and this will be the numbers that tend to be one or zero or something like that. Get the nice point x naught or y naught, let's label it. This is our p, and for us the x value that we're talking about here is 1.1. So let's just do one. The y value is .05, so a little nicer value would be one comma zero. Then we set up the equation. So we set up the equation for the linear approximation L of x, y. So remember how to do that? This is like little part A. We need z naught. Now z naught, remember, was F of x naught, y naught. And this is why we're using the nice point, because I don't know what these decimals are without a calculator. I can certainly plug in and get some nicer values, so this becomes one times the natural log of four times y, which is zero, four times zero plus one. And I guess I'll put the two out front again, so times two. Natural log of four times zero plus one is zero plus one, which is one. Natural log of one is in fact zero. And so I get basically zero times everything which is zero. So my z naught in this case is zero. Let's get some partial derivatives, so let's get the partial with respect to x of this function. I have two expressions here with an x, so we're going to need to do the product rule. And let's go carefully here. So first times derivative of the second logarithm is one over, so 4y This x, and then there's a times 1 coming from the coefficient on the x, so 2 times 1 over 4y plus x first times derivative of the second. Plus the 2nd logarithm of 4y plus x times the derivative of the first derivative 2x is just 2. And while we're getting partials, let's go ahead and get the partial with respect to y. This one is not a product rule, because the 2x in front now becomes a constant, so you get 2x in front. You get 1 over 4y plus x and then chain rule kicks in times that are the inside with respect to y and that is good old times 4. So that just turns out to be 8x over 4y plus x. Great, great, great. Now remember, this is the part that everybody forgets that we plug in. So we want f of x at our point p and we want f of y at our point p. Let's go ahead and be careful and remember p, is the nice point we're working with, we're working with a nice point this entire time, x is 1 and is 0. So when I do that, I get 2 times 1 and then times 1 over 0 plus 1 basically or nicely 1, such as 2 plus 2 times log of 0 plus 1, log of one is 0 and this becomes 2 plus 0, so that's 2, okay. And then the partial with respect to y I get 2 times 1, 1 over 4 times 0 plus 1, which is just 1 again and then times 8 times 4. So we have 8 over 1 which is 8. So we have 2 and 8, always plug in, always plug in, always plug in. Our last but not least, we are almost there. So we set up our equation. So these are multi steps so we have that the linear approximation. Appraised prize equal to so z now we set with zero. I'll write it in anyway. Plus f of x2. So that's coming from our value here, times x minus x naught, that's 1, x1 is 1 plus our partial derivative with respect to y plugged in, that's 8y minus 0. Clean all that up and you get 2 parentheses, x minus 1 plus 8y. Okay so big picture, was the point of all this. This is the equation of the tangent plane much nicer 2 times x minus 1 plus 8 times y and the point is near 1, 0. So as long as you if you wiggle 1 a little bit an wiggle 0 a little bit, that nicer expression 2x minus 1 which represents this tangent plane will give you the exact same thing. Has the value of this expression and so, let's try. So here comes the final piece. Let's find f of 1.1 comma 0, which is exactly what we wanted to do in the beginning. Remember, thinking this as f of 1.1 comma 0. And we know it's approximated by L of 1.1 comma 0. And here we go, we plug in, you get 2, 1.1 minus 1 plus 8 times 0. This just becomes 2 times point 1 or better known as point 2. Now is a good time to check and you can see how good we did. You can go ahead and plug this value into a calculator. So our final answer for the approximation is 0.2. And again if you check this against the original answer and leave, this is an exercise you will find out that it is pretty close, all things considered. So this is one way to get at numerical approximations. As you can tell, these are multi step. They're like a recipe. If you mess up Step 1, the whole thing is messed up. As you want to go very carefully through each point you want to check it multiple times. You want to watch out for any algebra mistakes or derivative mistakes and go really slowly, really careful through all these things. You'll do some practice problems. Test yourself so good you can be, but just go slow and with all these things always check your work. Great job in this video will see,
