Example The function f is defined by 2 f x y x xy 2 f a a 2 2 2 2 3 2 2 f a a a

Example the function f is defined by 2 f x y x xy 2 f

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Example: The function f is defined by ( , ) 2 f x y x xy = + = 2 ( , ) ? f a a = 2 2 2 2 3 ( , ) 2 2 f a a a a a a a = + = + 11.3 The graph of a function of two variables Suppose z = f ( x , y ) is a function of two variables over a domain D in the xy -plane. The graph of the function f is the set of all points ( x , y , f ( x , y )) obtained by letting ( x , y ) “run through” D . 11.3 Example The graph of z = x 2 + y 2 The surface is called a paraboloid. 11.1 Example Determine the domain of the function given by the following formula. 2 2 1 1 ( , ) 1 ln( 2) 1 f x y x x x y x y = - + + + - + - + + - + + - + - + Domain: 1 x 0 x y - - 2 2 1 0 x y + + + + 2 0 x - > - 11.2 Partial derivatives with two variables Consider the function The rate of change of z w.r.t. x is given by: The rate of change of z w.r.t. y is given by: Write 2 ln z x y = + 2 dz x dx = 1 dz dy y = and instead of z z x y and dz dz dx dx 11.1 Functions of two variables is called the partial derivative of z = f ( x , y ) w.r.t. x . You can also use the following notations: z x ' x z ' ( , ) x f x y 1 ' ( , ) f x y
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3 11.2 Example Find the partial derivatives of the following function: 3 ( , ) ln f x y x y xy y = + 3 2 ( ln ) 3 ln (holding constant) x y xy y x y y y x y + = + = 3 3 ( ln ) ln (holding constant) x y xy y x x y x y x + = + + + + + 11.8 Partial elasticities The definition of elasticity ( ) '( ) ( ) x x El f x f x f x = = If f is differentiable in x and f ( x ) 0 we define the elasticity of f w.r.t. x as: 1. Function of one variable: f(x) 2. Function of 2 variables: z=f(x,y)
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