ans eval alpha110 m1 k1 ans x2 t

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Unformatted text preview: u can draw and sum boxes evaluated at the right-hand side or at the midpoint of the box. Mixed Partial Derivatives This section describes the D operator for derivatives and gives an example of a function whose mixed partial derivatives are different. Consider the following function. > f := (x,y) -> x * y * (x^2-y^2) / (x^2+y^2); f := (x, y ) → x y (x2 − y 2 ) x2 + y 2 The function f is not defined at (0, 0). > f(0,0); Error, (in f) numeric exception: division by zero At (x, y ) = (r cos(θ), r sin(θ)) the function value is > f( r*cos(theta), r*sin(theta) ); r2 cos(θ) sin(θ) (r2 cos(θ)2 − r2 sin(θ)2 ) r2 cos(θ)2 + r2 sin(θ)2 As r tends to zero so does the function value. > Limit( %, r=0 ); 7.1 Introductory Calculus • 237 r2 cos(θ) sin(θ) (r2 cos(θ)2 − r2 sin(θ)2 ) r →0 r2 cos(θ)2 + r2 sin(θ)2 lim > value( % ); 0 Thus, you can extend f as a continuous function by defining it to be zero at (0, 0). > f(0,0) := 0; f(0, 0) := 0 The above assignment places an entry in f ’s remember table. Note: A remember table is a hash table in which the arguments to a procedure call are stored as the table index, and the result of the procedure call is stored as the table value. For more information, refer to the ?remember help page and the Maple Advanced Programming Guide. Here is the graph of f . > plot3d( f, -3..3, -3..3 ); The partial derivative of f with respect to its first parameter, x, is > fx := D[1](f); fx := (x, y ) → y (x2 − y 2 ) x2 y x2 y (x2 − y 2 ) +2 2 −2 x2 + y 2 x + y2 (x2 + y 2 )2 This formula does not hold at (0, 0). 238 • Chapter 7: Solving Calculus Problems > fx(0,0); Error, (in fx) numeric exception: division by zero Therefore, you must use the limit definition of the derivative. > fx(0,0) := limit( ( f(h,0) - f(0,0) )/h, h=0 ); fx(0, 0) := 0 At (x, y ) = (r cos(θ), r sin(θ)) the value of fx is > fx( r*cos(theta), r*sin(theta) ); r sin(θ) (r2 cos(θ)2 − r2 sin(θ)2 ) r3 cos(θ)2 sin(θ) +2 2 r2 cos(θ)2 + r2 sin(θ)2 r cos(θ)2 + r2 sin(θ)2 −2 r3 cos(θ)2 sin(θ) (r2 cos(θ)2 − r2 sin(θ)2 ) (r2 cos(θ)2 + r2 sin(θ)2 )2 > combine( % ); 3 1 r sin(3 θ) − r sin(5 θ) 4 4 As the distance r from (x, y ) to (0, 0) tends to zero, so does |f x(x, y ) − f x(0, 0)|. > Limit( abs( % - fx(0,0) ), r=0 ); r →0 1 3 lim − r sin(3 θ) + r sin(5 θ) 4 4 > value( % ); 0 Hence, f x is continuous at (0, 0). By symmetry, the same arguments apply to the derivative of f with respect to its second parameter, y . > fy := D[2](f); 7.1 Introductory Calculus • 239 fy := (x, y ) → x (x2 − y 2 ) x y2 x y 2 (x2 − y 2 ) −2 2 −2 x2 + y 2 x + y2 (x2 + y 2 )2 > fy(0,0) := limit( ( f(0,k) - f(0,0) )/k, k=0 ); fy(0, 0) := 0 Here is a mixed second derivative of f . > fxy := D[1,2](f); fxy := (x, y ) → −2 x2 − y 2 x2 x2 (x2 − y 2 ) +2 2 −2 x2 + y 2 x + y2 (x2 + y 2 )2 y2 y 2 (x2 − y 2 ) x2 y 2 (x2 − y 2 ) −2 +8 x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )3 Again, the formula does not hold at (0, 0). > fxy(0,0); Error, (in fxy) numeric exception: division by zero The limit definition is > Limit( ( fx(0,k) - fx(0,0) )/k, k=0 ); k→0 lim − 1 > fxy(0,0) := value( % ); fxy(0, 0) := −1 The other mixed second derivative is > fyx := D[2,1](f); fyx := (x, y ) → −2 x2 − y 2 x2 x2 (x2 − y 2 ) +2 2 −2 x2 + y...
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This note was uploaded on 08/27/2012 for the course MATH 1100 taught by Professor Nil during the Spring '12 term at National University of Singapore.

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