2010-03-29 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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Unformatted text preview: MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review: Functions of several variables I: Partial derivatives 1 2. Function of several variables II: Vector-valued functions 2 2.1. Introductory example 2 2.2. Definition 2 2.3. Linear approximation and the Jacobian matrix 4 1. Review: Functions of several variables I: Partial derivatives Definition of partial derivative of functions of two independent variables. Suppose that f is a function of two independent variables x and y . The partial derivative of f with respect to x is defined by f ( x,y ) x = lim h f ( x + h,y )- f ( x,y ) h The partial derivative of f with respect to y is defined by f ( x,y ) y = lim h f ( x,y + h )- f ( x,y ) h Tangent plane Let f ( x,y ) be a real-valued function of two variables. If the tangent plane to the graph of f at the point ( x ,y ,z ) = ( x ,y ,f ( x ,y )) exists, then it is given by the equation z- z = f ( x ,y ) x ( x- x ) + f ( x ,y ) y ( y- y ) The linear approximation The linear approximation of a function f ( x,y ) near a point ( x ,y ) is given by L ( x,y ) = f ( x ,y ) + f ( x ,y ) x ( x- x ) + f ( x ,y ) y ( y- y ) = f ( x ,y ) + f ( x ,y ) x , f ( x ,y ) y x- x y- y . provided the function is differentiable. Date : 2010-03-29. 1 2 JING LI 2. Function of several variables II: Vector-valued functions 2.1. Introductory example. So far, we have only considered real-valued functions, f : R n R ( x 1 ,x 2 ,...,x n ) f ( x 1 ,x 2 ,...,x n ) Now, we want to be able to describe several quantities that are all dependent on the same variables.Now, we want to be able to describe several quantities that are all dependent on the same variables....
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2010-03-29 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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