Engineering Calculus Notes 369

Engineering Calculus Notes 369 - f (0 . 8 , 1 . 9); does...

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3.7. HIGHER DERIVATIVES 357 (i) f ( x,y ) = xy x 2 + y 2 2. Find all second-order derivatives of each function below: (a) f ( x,y,z ) = x 2 + y 2 + z 2 (b) f ( x,y,z ) = xyz (c) f ( x,y,z ) = xyz (d) f ( x,y,z ) = 1 x 2 + y 2 + z 2 (e) f ( x,y,z ) = e x 2 + y 2 + z 2 (f) f ( x,y,z ) = xyz x 2 + y 2 + z 2 3. Find the degree two Taylor polynomial of the given function at the given point: (a) f ( x,y ) = x 3 y 2 at ( 1 , 2). (b) f ( x,y ) = x y at (2 , 3). (c) f ( x,y,z ) = xy z at (2 , 3 , 5). 4. Let f ( x,y ) = xy x + y . (a) Calculate an approximation to f (0 . 8 , 1 . 9) using the degree one Taylor polynomial at (1 , 2), T (1 , 2) f ( x, y ). (b) Calculate an approximation to f (0 . 8 , 1 . 9) using the degree two Taylor polynomial at (1 , 2), T 2 f ((1 , 2)) x , y . (c) Compare the two to the calculator value of
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Unformatted text preview: f (0 . 8 , 1 . 9); does the second approximation improve the accuracy, and by how much? Theory problems: 5. (a) Show that for any C 2 function f ( x,y,z ), ∂ 2 f ∂x∂z ( x ,y ,z ) = ∂ 2 f ∂z∂x ( x ,y ,z ) . (b) How many distinct partial derivatives of order 2 can a C 2 function of n variables have? 6. (a) Show that for any C 3 function f ( x,y ), ∂ 3 f ∂x∂y∂x ( x ,y ) = ∂ 3 f ∂y∂ 2 x ( x ,y ) ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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