Now, we'll discuss the quality of linear approximation of the function of many variables. It's the same as the quality of tangent plane. So, if we draw the tangent plane to the surface which is the graph of the function, is it true that the points which belong to the tangent plane, they are close to the points which belong to the surface? This is not necessarily always true and I'll explain with such an example. Let's consider a function Z, which is the cubic root of the sum X cubed plus Y cubed. The question is, draw the tangent plane to the graph of this function at the origin. So, we are interested in drawing a tangent plane at zero, zero. In order to find tangent plane, we know we need to calculate both the partial derivatives at the origin. So we start with Z X at zero, zero. We substitute zero value for Y, take the cubic root of X cubed which is X, so we get one then, and another derivative is also one. Both values enable us to write down equation at tangent plane easily. So the Z value at the origin is zero, so then we have Z equals- by the way X 0, Y 0 are zeros, so we have simply X plus Y, and it looks like a tangent plane but, we need to find out whether the quality of approximation is good enough. What I mean is the following. If we compare two values, the original function, the value calculated at a point which lies in the neighborhood or the origin. That means small values of X and Y, and this is a linear function. So let's compare two values by subtracting the corresponding Z values. So, how good is this approximation, linear approximation or this function in the neighborhood of the zero? In order to check the quality of approximation, let's introduce the distance from a point with coordinates X and Y, and the origin is given by the square root of the sum of two squares. I indicate this distance from a point. So this is X axis, Y axis, this is the point, and this is the distance to the origin. I'm using Greek letter Rho. So, the quality of linear approximation of a given function will be good enough if as X and Y tend to zero, the difference between the values of two functions, the original function Z and its linear approximation, tends to zero quicker than Rho. How to write it down. If we subtract from the value of the function, it's linear approximation and divide this difference by Rho, as X and Y tends to zero. This approximation is good enough if this limit exists and equals zero, that means that this difference tends to zero faster than the distance between the point and the origin. So, we need to check for our particular function whether it's true or not. The easiest way to investigate the existence of this limit would be to turn to polar coordinates. So, in Cartesian plane, the correspondence is between X and Y and the polar coordinates given by a pair of variables, R and Phi. Although, we have introduced the distance between the point and the origin using Rho letters. So, in this case, I'll be using instead of R, Rho. Then X will be Rho cosine Phi and Y will be Rho sine Phi. Okay, so what we do, we substitute these formulas into the quotient. Then, we need to check whether such a limit exists, as Rho tends to zero. So we have R, the cubic root of the sum of two cubes, minus, minus and this is divided by Rho. So, Rho is cancelled out and we have a function which depends only on trigonometric functions. Well, clearly such a limit doesn't exist because in this formula we have no Rho and when we approach the origin using different straight lines. For example, if I choose a particular value for Phi. If Phi equals zero then, what do I get? Sine times zero here and there the cosine is left, the cosine is one. So I get zero, but if I choose some other value for Phi, let's suppose, I'm taking another value like Pi over four then, I get something else. Well, clearly this is not zero, we can check. So that means that such a limit, simply doesn't exist. What will be the conclusion drawn from this calculation? Not necessarily, but we get a good linear approximation or even in the case when both partial derivatives exist. But, there is an important theoretical fact that whenever we have a function of two variables, actually this result is also valid for a dimensional case, but let's consider just X and Y as independent variables. So, given, a function Z f(x,y) which belongs to C one class, then, the limit as Rho tends to zero of the quotient when we subtract from delta F change in the value of the function, its total differential divided by Rho is zero. So that means the tanget plane, really the points which belong to the tangent plane are very close to the points which belong to the surface, which is the graph of the function and the quality of linear approximation is also very high.