Tangent Plane. Nobuyuki TOSE. October 02, Nobuyuki TOSE. Tangent Plane

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1 October 02, 2017

2 The Equation of a plane Given a plane α passing through P 0 perpendicular to n( 0). For any point P on α, we have n PP 0 = 0 When P 0 has the coordinates (x 0, y 0, z 0 ), P 0 (x, y, z) and p n = q r the above identity is expresses using the coordinates by p(x x 0 ) + q(y y 0 ) + r(z z 0 ) = 0

3 The Equation of a plane

4 In case r = 0 In case r = 0, the vector n is parallel to x y plane. Then the plane α is orthogonal (perpendicular) to x y plane. r = 0 n x yplane, α x y plane

5 Partial Derivative (Review) Given a function of two variable z = f (x, y) We define a function in x F(x) := f (x, b) to define f x (a, b) = F (a) y =b

6 We find the tangent plane of the graph z = f (x, y) at (a, b, f (a, b)). We find the coefficients A and B of the equation z = A(x a) + B(y b) + f (a, b) (1) y =b

7 (2) The tangent plane intersected with the cross-section y = b is the tangent line of z = F(x) at x = a. Moreover the equation (1) with y assigned to be b tunrs to z = A(x a) + f (a, b) Accordingly the slope of the line on the cross-section y = b is We can also show A = F (a) = f x (a, b) B = f y (a, b) Consequently the tangent plane is expressed by the equation z = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b)

8 Examples Consider the function z = f (x, y) = 4x 3 4 y 1 4 at (x, y) = (a, b) = (10 4, 625). The the partial derivatives of f are and we find f x (x, y) = 3x 1 4 y 1 4, fy (x, y) = x 3 4 y 3 4 f x (10 4, 625) = 1.5, f y (10 4, 625) = 8 Accordingly the tangent plane at (x, y) = (10 4, 625) is z = 1.5(x 10 4 ) + 8(y 625)

9 Marginal Products We consider a production function Q = f (K, L) of Capital input K and Labor input L. Then Q = f (K 0 + K, L 0 ) f (K 0, L 0 ) f K (K 0, L 0 ) K In this situation F K (K 0, L 0 ) is the Marginal Product of Capital (MPK) at (K, L) = (K 0, L 0 ).

10 Marginal Products We consider the Cobb-Douglas production function Q = F(K, L) = 4K 3 4 L 1 4 around (K, L) = (K 0, L 0 ) = (10 4, 625). Then the Marginal Prodct of Capital (MPK) and the Marginal Product of Labor (MPL) at (K 0, L 0 ) are MPK = F K (K 0, L 0 ) = = 1.5 MPL = F K (K 0, L 0 ) = = 8 We estimate Q = F( , 625) = 20, by F( , 625) F(10 4, 625)+F K (10 4, 625) 100 = 20, =

11 Tangent Line of a Curve Assume a curve C is given by a function of two variables. g(x, y) = 0 For example, the unit circle is given by g(x, y) := x 2 + y 2 1 = 0 We are given a point P 0 (a, b) on the curve C and we are to find the equation of the tangent line of C at P 0 (a, b).

12 Tangent Line of a Curve We consider the tangent plane of at (a, b, 0): z = g(x, y) z = g x (a, b) (x a) + g y (a, b) (y b) (2) The intersection of the tangent plane with x y plane is given by g x (a, b) (x a) + g y (a, b) (y b) = 0 This is the equation for the tangent line.

13 Gradient Vector The equation (2) is interpreted as ( ) gx (a, b) g y (a, b) ( ) x a = 0 y b This means that the voctor (g)(a, b) := ( ) gx (a, b) g y (a, b) is perpendicular to the tangent line. (g)(a, b) is called the gradient vector of g at (a, b).

14 Example We consider the unit circle g(x, y) := x 2 + y 2 1 = 0 The partial derivative of g is g x = 2x, g y = 2y Accordingly the tangent line at the point (a, b) on the UC is 2a(x a) + 2b(y b) = 0

15 Derivative of an implicit function Assume that the curve C gives rise to a function y = ϕ(x) in a neighborhood of (a, b). Then if g y (a, b) 0, the tangent line at (a, b) is y = g x(a, b) (x a) + b g y (a, b) and we get ϕ (a) = g x(a, b) g y (a, b) For example, we consider the curve g(x, y) = x 2 + y 2 1 = 0 at (a, b) with b > 0. Then the curve is explicitly written by y = ϕ(x) = 1 x 2 and we find ϕ (a) = 2a 2b = a b

16 Rate of Marginal Substitution (MRS) A consumer purchases two goods A and B with x units and y units of quantities. The Utility of the consumer is given by u(x, y). The curve u(x, y) = u(a, b) is called the Indifference Curve through (a, b). Then the Marginal Rate of Substitution at (a, b) is defined by MRS = u x(a, b) u y (a, b). In case the amount to purchase A is increased from a by x, then the consumer should decrease the amount to purchase B approximately by MRS x in order to keep the utility.

CHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.

April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as