week8 (3).pdf - Week 8 Functions of several variables(part...

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Week 8: Functions of several variables (part 3) Dr Jorge Vit´ oria School of Engineering and Mathematical Sciences Department of Mathematics City, University of London March 27, 2018 Implicit differentiation in several variables. Last lecture’s considerations on implicit differentiation can be extended to more than two variables, where one of these variables is considered as a function in all other variables. More generally, let f be a function in n + 1 variables x 1 , x 2 , ..., x n , y . Consider y as a variable in the remaining variables; that is, y = y ( x 1 , x 2 , .., x n ). Consider the level set consisting of all points ( x 1 , x 2 , .., x n , y ) satisfying f ( x 1 , x 2 , ...x n , y ) = c , where c is some constant. Dr Jorge Vit´oria Mathematics for Economics (Post A level) In order to calculate the partial derivatives ∂y ∂x i , we use the chain rule together with the observation ∂x i ∂x i = 1 , ∂x i ∂x j = 0 , for 1 i, j n and i 6 = j . Provided that ∂f ∂y 6 = 0 we get as in the case of two variables that ∂y ∂x i = - ∂f ∂x i ∂f ∂y Dr Jorge Vit´oria Mathematics for Economics (Post A level) Example. Consider the function f ( x 1 , x 2 , y ) = x 1 - 2 x 2 - 3 y + y 2 and the level curve f ( x 1 , x 2 , y ) = - 2. ∂y ∂x 1 = - 1 - 3 + 2 y = 1 3 - 2 y , ∂y ∂x 2 = - - 2 - 3 + 2 y = 2 2 y - 3 . These are defined except if y = 3 / 2. Dr Jorge Vit´oria Mathematics for Economics (Post A level) Linear Approximations: one variable Definition. The linear approximation to f at x = a is the tangent line to the graph of f at the point ( a, f ( a )). This tangent line is given by the equation y = f ( a ) + f 0 ( a )( x - a ) When x close to a this approximates the value of the function f ( x ). For examples of use in EMEA see section 7.4. Dr Jorge Vit´oria Mathematics for Economics (Post A level) If we set b = f ( a ), then the tangent line equation can be rewritten in the form y - b = f 0 ( a )( x - a ) . Since y - b approximates the difference in value of the function f as x varies around a , one finds in the literature the notation dy = f 0 ( a ) dx , or also Δ y = f 0 ( a x . Dr Jorge Vit´oria Mathematics for Economics (Post A level) Example. Suppose C ( x ) is the cost of produce x units. Suppose that C 0 (50) = 2. This information can be interpreted as follows: Producing 51 units rather than 50 will cost approximately and extra 2; that is, C (51) C (50) + 2. This is useful for evaluating the marginal cost of a product. Dr Jorge Vit´oria Mathematics for Economics (Post A level) Linear approximation: two variables A function z = f ( x, y ) in two variables can be represented as a surface in the 3-dimensional space. If f is linear, that is of the form z = ax + by + c for some constants a , b , c , then this surface is a plane. Dr Jorge Vit´oria Mathematics for Economics (Post A level)
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Definition. The linear approximation of a function f ( x, y ) at a point ( a, b ) is the tangent plane to the graph of f at the point ( a, b, f ( a, b )). This tangent plane is the set of all points ( x, y, z ) satisfying the formula z = f ( a, b ) + f x ( a, b )( x - a ) + f y ( a, b )( y - b ) , where f x = ∂f ∂x and f y = ∂f ∂y .
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