UNIT7 - Unit 7 Partial Derivatives and Optimization We have...

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Unit 7 Partial Derivatives and Optimization We have learned some important applications of the ordinary derivative in Fnding maxima and minima. We now move on to a topic called partial derivatives which may be used to Fnd local maxima and minima of surfaces ( functions of two variables). Defnition 7.1. Given some function y = f ( x )thenw ehav etwoway so f interpreting the derivative dy dx (1) Intuitively , dy dx measures the rate of change of the variable y with respect to the variable x (2) Geometrically , dy dx can be interpreted as the equation for the slope of the tangent line to the curve y = f ( x ). Similarily, if we have z = f ( x, y ), a function of two variables we can deFne what is called a partial derivative : ∂f ∂x = ∂z :Th e Partial derivative of z with respect to x . and ∂y = :The Partial derivative of z with respect to y . Partial derivatives ( or simply partials ) can be interpreted geometrically as well as intuitively. More on that later, now let’s learn how to calculate partials. To calculate we simply treat y as if it were a constant and then take the usual derivative of f(x,y) with respect to x . Similarily, to calculate we simply treat x as if it were a constant and then take the usual derivative of f(x,y) with respect to y . Recalling that x and y are just “dummy variables”, the generalization of this process is transparent. 1
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Example 1. Given the function z =2 xy 3 +3 x 2 +2 y 3 , calculate each of its partials. Solution: To Fnd ∂z ∂x we treat y as if it were a constant and then di±erentiate so: (2 xy 3 x 2 y 3 )=2 y 3 +6 x +0=2 y 3 x To Fnd ∂y we treat x as if it were a constant and then di±erentiate ( with respect to y) so: (2 xy 3 x 2 y 3 x (3 y 2 )+0+6 y 2 =6 xy 2 y 2 y 2 ( x +1) Example 2. given the function f ( x, y, z )= e xy + xyz , calculate each of its partials. Solution: To Fnd ∂f we treat y and z as if they were constant and then di±erentiate so: ( e xy + xyz e xy ( ( xy )) + yz = e xy y + Similarily, in Fnding we treat x and z as if they were constant and then di±erentiate ( with respect to y) so: ( e xy + xyz e xy ( ( xy )) + xz = e xy x + xz And, in Fnding we treat x and y as if they were constant and then di±er- entiate ( with respect to z) so: ( e xy + xyz e xy ( ( xy )) + xy = e xy (0) + xz = xz Along with these new deFnitions comes some new notation which paral- lels that used for functions of single variables: ²unctions of one variable, say f(x) ²unctions of 2 variables, say f(x,y) f 0 ( x ) or df dx or f x or f x ( x, y ) f 0 ( a ) or ( dx )] x = a ( )] x = a or f x ( a, y ) ( )] x = a,y = b or f x ( a, b ) 2
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The following example is very important, it illustrates and adds to the previous points and should be read carefully. Example 3. Suppose f ( x, y )= x 2 y + xe xy + x 3 4 , Fnd (a) f y ( b ) f x ( x, 0) ( c ) f x (0 , 1) ( d ) f y (1 , 1) Solution: (a) To calculate f y we simply hold x as constant and di±erentiate with re- spect to y: f y = ∂y ( x 2 y + xe xy + x 3 4 ) =2 xy + xe xy ( xy )+ ( x 3 4 ) xy + xe xy ( x )+0 xy + x 2 e xy So we have f y xy + x 2 e xy (b) To calculate f
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This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.

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UNIT7 - Unit 7 Partial Derivatives and Optimization We have...

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