Linear and quadratic Taylor polynomials for functions of several variables.


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1 ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is one of the most important applications of differential calculus to economics. In general, there are two steps to this process: (i) finding the the points where extreme values may occur, (ii) analyzing the behavior of the function near these points, to determine whether or not extreme values actually occur there. For a function of one variable, we saw that step (i) consisted of finding the critical points of the function, step (ii) consisted of using the first or second derivative test to analyze the behavior of the function near the point. We use the same two steps to find the critical points classify the critical values of functions of several variables, to underst how the procedure generalizes, we need to first underst the linear quadratic Taylor s polynomials for functions of several variables, the approximations that they provide. Comment: I will assume throughout this note that all the functions being discussed have continuous first, second third order derivatives (or partial derivatives). This assumption justifies the claims made below about the accuracy of the approximations. 1. The onevariable case. To makes sense of Taylor polynomials in several variables, we first recall what they look like for functions of one variable. 1.1 The first order (linear) approximation The first order Taylor polynomial for y = f(t), centered at t 0 is the linear function With this definition of T 1, the approximation is very good if t is sufficiently close to t 0. T 1 (t) = f(t 0 ) + f (t 0 ) (t t 0 ). (1) f(t) T 1 (t), () This approximation is also called the tangent line approximation because the graph y = T 1 (t) is the tangent line to the graph y = f(t) at the point (t 0, f(t 0 )). 1. The second order (quadratic) approximation The second order Taylor polynomial for the function y = f(t), centered at t 0 is the quadratic function T (t) = f(t 0 ) + f (t 0 )(t t 0 ) + f (t 0 ) (t t 0 ). () For a more thorough discussion of Taylor polynomials for functions of one variable, please see SN 7 on the review page of the 11A website: yorik/11a/review.htm. The error of approximation f(t) T 1 (t) is less than a multiple of t t 0. As t approaches t 0, the squared difference t t 0 goes to 0 very rapidly. 1
2 You should note that T (t) = T 1 (t)+ f (t 0 ) (t t 0 ). As for the linear approximation, the quadratic approximation f(t) T (t). (4) is very good if t is sufficiently close to t 0. In fact, the quadratic approximation is typically much better than the linear approximation once t is very close to t 0. At this point the key question is: Why do the Taylor polynomials T 1 (t) T (t) do such a good job of approximating the original function f(t) in the neighborhood of t 0? The answer is provided by looking at what happens at the point t 0 itself. The function T 1 (t) is the unique linear function that satisfies the conditions T 1 (t 0 ) = f(t 0 ) T 1(t 0 ) = f (t 0 ). (5) That is, T 1 (t) has the same value as f(t) at t 0 T 1 (t) is changing at the same rate as f(t) at t 0. Likewise, the function T (t) is the unique quadratic function that has the same value at t 0 as f(t), has the same rate of change at t 0 as f(t) has the same concavity at t 0 as f(t) (T (t) is changing at the same rate as f (t) at the point t 0 ). I.e., T (t 0 ) = f(t 0 ), T (t 0 ) = f (t 0 ) T (t 0 ) = f (t 0 ). (6) In other words, the functions T 1 (t) T (t) behave very much like f(t) at the point t 0, this similar behavior at t 0 carries over to points that are close to t 0. To find analogous polynomial approximations for a function of several variables, we will impose the same sort of conditions on the values of the polynomial on the values of the partial derivatives of the polynomial at a specified point. This will give simple equations for the coefficients of the polynomial, as I will describe in the sections that follow.. The linear Taylor polynomial in two variables. Suppose that we want to find a linear polynomial in two variables that approximates the (differentiable) function F (x, y) in the neighborhood of the point (x 0, y 0 ). That is, we want to find a function T 1 (x, y) = A + B(x x 0 ) + C(y y 0 ) such that T 1 (x, y) F (x, y) if (x, y) is close to (x 0, y 0 ). We choose to write T 1 (x, y) as a function of (x x 0 ) (y y 0 ) (instead of the more traditional T 1 (x, y) = a + bx + cy) because it emphasizes the role of the point (x 0, y 0 ) in the approximation it makes the equations we will solve much simpler. The comments at the end of the previous section lead to the idea of imposing conditions on the values of T 1 (x, y), T 1x (x, y) T 1y (x, y) at the point (x 0, y 0 ). In analogy with the properties of T 1 (t) described in (5), we require that T 1 (x 0, y 0 ) = F (x 0, y 0 ), T 1 x (x 0, y 0 ) = F x (x 0, y 0 ) T 1 y (x 0, y 0 ) = F y (x 0, y 0 ). If t t 0 is small, then f(t) T (t) is less than a multiple of t t 0. Thus, if t t 0 < 1, then both t t 0 t t 0 are very small t t 0 < t t 0. This implies (with more work) that once t t 0 is small enough, f(t) T (t) < f(t) T 1 (t).
3 These conditions give very simple equations for the unknown coefficients A, B C. Specifically, since T 1 (x 0, y 0 ) = A, it follows that A = F (x 0, y 0 ); since T 1x = B, it follows that B = F x (x 0, y 0 ); since T 1y = C, it follows that C = F y (x 0, y 0 ). In other words, the first order Taylor polynomial for F (x, y), centered at (x 0, y 0 ) is given by T 1 (x, y) = F (x 0, y 0 ) + F x (x 0, y 0 ) (x x 0 ) + F y (x 0, y 0 ) (y y 0 ). (7) Using more advanced techniques, it possible to show that the approximation is very good if (x, y) is close to (x 0, y 0 ). Example 1. so F (x, y) T 1 (x, y) If F (x, y) = 5x + y (x 0, y 0 ) = (1, 10), then F x = 5 (5x + y) 1/ F y = (5x + y) 1/, F (1, 10) = 5 = 5, F x (1, 10) = 5 10 = 1 F y(1, 10) = 1 5. It follows that the linear Taylor approximation to 5x + y, centered at (1, 10), is given by T 1 (x, y) = (x 1) + 1 (y 10). 5 To test the accuracy of the approximation that this provides, I ll use it to estimate 7.04 = 5 (1.) + (10.5) = F (1., 10.5). Plugging (x, y) = (1., 10.5) into T 1 (x, y), we have T 1 (1., 10.5) = = 5.04, 5 giving the estimate 7.04 = F (1., 10.5) T1 (1., 10.5) = This estimate is exactly 1/50 away from the truth, since 7.04 = = 1/50.. The quadratic Taylor polynomial in two variables. Two find the formula of the quadratic Taylor approximation for the function F (x, y), centered at the point (x 0, y 0 ), we repeat the procedure we followed above for the linear polynomial, but we take it one step further. In analogy with the conditions satisfied by T (t) in the onevariable setting (shown in (6)), we want to find the coefficients A, B, C, D, E G of the quadratic function T (x, y) = A + B(x x 0 ) + C(y y 0 ) + D(x x 0 ) + E(y y 0 ) + G(x x 0 )(y y 0 ) These techniques are outside the scope of this course, but well within the realm of an upper division math class.
4 that satisfies the conditions T (x 0, y 0 ) = F (x 0, y 0 ), T x (x 0, y 0 ) = F x (x 0, y 0 ), T y (x 0, y 0 ) = F y (x 0, y 0 ), T x (x 0, y 0 ) = F x (x 0, y 0 ), T y (x 0, y 0 ) = F y (x 0, y 0 ), T x y (x 0, y 0 ) = F x y (x 0, y 0 ). In other words, we want T (x, y), its first order partial derivatives its second order partial derivatives to all have the same values as the corresponding derivatives of F (x, y) at the point (x 0, y 0 ). The values of the partial derivatives of T (x, y) at the point (x 0, y 0 ) are all very simple expressions in the coefficients of T (x, y). Indeed, we have T (x 0, y 0 ) = A, T x (x 0, y 0 ) = B, T y (x 0, y 0 ) = C, T x (x 0, y 0 ) = D, T y (x 0, y 0 ) = E, T x y (x 0, y 0 ) = G, as you should verify by doing the calculations yourself. Comparing the two lists above, we find that the quadratic Taylor polynomial for F (x, y), centered at (x 0, y 0 ), is given by T (x, y) = F (x 0, y 0 ) + F x (x 0, y 0 )(x x 0 ) + F y (x 0, y 0 )(y y 0 ) + F xx(x 0, y 0 ) (x x 0 ) + F yy(x 0, y 0 ) (y y 0 ) + F xy (x 0, y 0 )(x x 0 )(y y 0 ). (8) As in the onevariable case, the linear constant coefficients of T (x, y) are the same as those of T 1 (x, y). In other words, we have T (x, y) = T 1 (x, y) + F xx(x 0, y 0 ) (x x 0 ) + F yy(x 0, y 0 ) (y y 0 ) + F xy (x 0, y 0 )(x x 0 )(y y 0 ). Also as in the onevariable case, the quadratic terms in T (x, y) tend to make the approximation F (x, y) T (x, y) more accurate than the approximation F (x, y) T 1 (x, y), when (x, y) is very close to (x 0, y 0 ). 4
5 Example. Let s see if the quadratic Taylor polynomial gives a more accurate approximation than the linear one for the function points in Example 1. We already have the constant linear coefficients of T (why?), so it remains to find the quadratic coefficients. First we compute the second order partial derivatives of F (x, y) = 5x + y: F xx = 5 4 (5x + y) /, F yy = (5x + y) / F xy = 5 (5x + y) /. Next, we evaluate these at the point (x 0, y 0 ) = (1, 10), remembering that 5 / = 1 15 : F xx (1, 10) = 1 0, F yy(1, 10) = 1 15 F xy(1, 10) = Using these numbers the ones we found in Example 1 in Equation (8), we find that T (x, y) = (x 1) (y 10) 5 40 (x 1) 1 50 (y 10) 1 (x 1)(y 10), 50 T (1., 10.5) = (0.) (0.5) (0.)(0.5) = This give the estimate 7.04 = F (1., 10.5) T (1., 10.5) = , which is closer to the correct value ( 7.04 = 5.) than the previous estimate, since ( T (1., 10.5) F (1., 10.5) = = 101 ) This is almost 5 times smaller than the error of approximation we had when we used the linear approximation. 4. Linear quadratic Taylor polynomials for functions of three variables. As the number of variables grows, the number of terms in both the linear quadratic Taylor polynomials grows as well, but the forms of the polynomials follow the same patterns that we saw in (7) (8) for the twovariable case. The formulas for a function of three variables appear below. 4.1 The linear Taylor polynomial for a function of three variables. The linear Taylor polynomial for the function F (x, y, z), centered at the point (x 0, y 0, z 0 ), has the form T 1 (x, y, z) = F (x 0, y 0, z 0 ) + F x (x 0, y 0, z 0 )(x x 0 ) + F y (x 0, y 0, z 0 )(y y 0 ) + F z (x 0, y 0, z 0 )(z z 0 ) As in the one twovariable cases, T 1 (x 1,..., x n ) satisfies the conditions T 1 (x 0, y 0, z 0 ) = F (x 0, y 0, z 0 ), T 1x (x 0, y 0, z 0 ) = F x (x 0, y 0, z 0 ), T 1y (x 0, y 0, z 0 ) = F y (x 0, y 0, z 0 ), T 1z (x 0, y 0, z 0 ) = F z (x 0, y 0, z 0 ). (9) 5
6 Indeed, as before, it is by requiring that these conditions be satisfied that the coefficients were found. Also as before, the approximation F (x, y, z) T 1 (x, y, z) is very accurate when (x, y, z) is sufficiently close to (x 0, y 0, z 0 ). 4. The quadratic Taylor polynomial for a function of three variables. The linear Taylor polynomial for the function F (x, y, z), centered at the point (x 0, y 0, z 0 ), has the form T (x, y, z) = F (x 0, y 0, z 0 ) + F x (x 0, y 0, z 0 )(x x 0 ) + F y (x 0, y 0, z 0 )(y y 0 ) + F z (x 0, y 0, z 0 )(z z 0 ) + F xx(x 0, y 0, z 0 ) (x x 0 ) + F yy(x 0, y 0, z 0 ) (y y 0 ) + F zz(x 0, y 0, z 0 ) (z z 0 ) + F xy (x 0, y 0, z 0 )(x x 0 )(y y 0 ) + F xz (x 0, y 0, z 0 )(x x 0 )(z z 0 ) + F yz (x 0, y 0, z 0 )(y y 0 )(z z 0 ) (10) The coefficients of T (x, y, z) are chosen so that the value of T those of all of its first second order partial derivatives are equal to the values of the corresponding partial derivatives of F (x, y, z). These conditions ensure, as before, that the approximation F (x, y, z) T (x, y, z) is very accurate if (x, y, z) is sufficiently close to (x 0, y 0, z 0 ). 5. Another example. The example in this section is meant to illustrate how the approximation a function of two variables by its quadratic Taylor polynomial is used in the classification of local extreme values. In particular you will see that we don t use the approximation to estimate specific values, but rather we use the Taylor polynomial to approximate the overall behavior of the function in the neighborhood of a critical point. This idea eventually leads to the generalization of the second derivative test to the case of functions of two or more variables. Consider the function f(x, y) = ) (1 x y e 0.75x +y, whose graph in the neighborhood of (0, 0) appears in Figure 1, below. The point (0, 0, 1) on the graph is the point from which the vertical arrow is extending, it appears to be the lowest point on the graph in its own immediate vicinity. In other words, it appears that f(0, 0) = 1 is a local minimum value of the function, meaning that f(0, 0) < f(x, y) for all points (x, y) that are sufficiently close to (0, 0). However pictures can be misleading, so it is important to have an analytic argument that shows that f(0, 0) is indeed a local minimum, without reliance on pictures. 6
7 ( ) Figure 1: The graph of f(x, y) = 1 x y e 0.75x +y. Analyzing the function f(x, y) as it is defined may be a little complicated, this is where the Taylor approximation of f(x, y), centered at (0, 0), becomes very useful. Applying the Taylor approximation, f(x, y) T (x, y), in this case, yields f(x, y) 1 + f x (0, 0)(x 0) + f y (0, 0)(y 0) + f xx(0, 0) (x 0) + f yy(0, 0) (y 0) + f xy (0, 0)(x 0)(y 0) = 1 + f x (0, 0) x + f y (0, 0) y + f xx(0, 0) x + f yy(0, 0) y + f xy (0, 0) xy, (which is accurate if (x, y) is sufficiently close to (0, 0)). The expression in the third row is fairly simple already, it will become simpler still once we evaluate all the partial derivatives of f(x, y) at (0, 0) compute the (numerical) coefficients. Differentiating once yields f x (x, y) = x +y + 1.5x (1 x e0.75x f y (x, y) = y e 0.75x +y + y 7 y ) e 0.75x +y ) (1 x y e 0.75x +y, (11)
8 evaluating these expressions at the point (0, 0) gives f x (0, 0) = 0 f y (0, 0) = 0, which means that T (x, y) has no linear terms in this instance. Differentiating the first derivatives ( collecting terms) yields the second order derivatives: f xx (x, y) = [ ( x + 4.5x ) + ( x ) ( )] 1 x y e 0.75x +y, [ )] f xy (x, y) = x y 1.5xy + xy (1 x y e 0.75x +y f yy (x, y) = [ ( y + 4y ) ) + ( + 4y ) ( )] 1 x y e 0.75x +y. As complicated as they appear, the second order partial derivatives are very easy to evaluate at the point (0, 0). Indeed f xx (0, 0) = [ 0 + ( ) (1 0 0)] e 0 = = 1.5, f xy (0, 0) = [ (1 0 0)] e 0 = 0 f yy (0, 0) = [ 0 + ( + 0) (1 0 0)] e 0 = 1 1 =. Returning to the Taylor approximation (11), inserting the values of the coefficients, we see that if (x, y) is close to (0, 0), then f(x, y) 1 + x 4 + y. This means that if (x, y) is close (but not equal to) to (0, 0), then f(x, y) f(0, 0) x 4 + y > 0. In other words, if (x, y) is sufficiently close to (0, 0), then f(x, y) f(0, 0), showing that f(0, 0) is a local minimum value, as we suspected. The conditions f x(x 0, y 0 ) = 0 f y(x 0, y 0 ) = 0 make (x 0, y 0 ) a critical point of the function f(x, y), as you might expect. More on that in SN. I recommend that you compute these partial derivatives on your own, for practice. 8
9 Exercises 1. Compute T (x, y) for the function centered at the point (x 0, y 0 ) = (1, 1). f(x, y) = x y + x y xy + x 5y,. Compute T (K, L) for the function Q(K, L) = 10K / L 1/, centered at the point (K 0, L 0 ) = (1, 8). Use the Taylor polynomial that you found to calculate the approximate value of Q(1.5, 8.4).. Compute T (u, v) for the function g(u, v) = (u + v)e u +v, centered at the point (u 0, v 0 ) = (0, 0). 4. Compute T (x, y) for the function H(x, y) = x + 5y, centered at the point (x 0, y 0 ) = (, ). Use the Taylor polynomial that you found to calculate the approximate value of H(.5,.1) = Compute T (x, y, z) for the function centered at the point (x 0, y 0, z 0 ) = (4, 5, 0.1). g(x, y, z) = ln x + 4 ln y z(5x + 8y 60), 9
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