Explanation Given information: The function is f ( x , y ) = tan − 1 x y Concept and Formula Used: Product rule ∂ ∂ y ( u ⋅ v ) = u ∂ v ∂ y + v ∂ u ∂ y ∂ ∂ y y n = n y n − 1 Calculation: It is given that f ( x , y ) = tan − 1 x y To find f x x we differentiate the function first with respect to x and then again with respect to x keeping y as constant. f x = ∂ ∂ x tan − 1 x y Using ∂ ∂ x tan − 1 x = 1 1 + x 2 f x = 1 1 + ( x y ) 2 ∂ ∂ x ( x y ) f x = 1 1 + ( x y ) 2 ⋅ y ∂ x ∂ x f x = y 1 + ( x y ) 2 = y 1 + x 2 y 2 Again differentiate with respect to x with y as constant ∂ ∂ x f x = ∂ ∂ x ( y 1 + x 2 y 2 ) using quotient rule ∂ ∂ x ( u v ) = v ∂ u ∂ x − u ∂ v ∂ x v 2 f x x = ( 1 + x 2 y 2 ) ∂ ∂ x y − y ∂ ∂ x ( 1 + x 2 y 2 ) ( 1 + x 2 y 2 ) 2 f x x = ( 1 + x 2 y 2 ) ⋅ 0 − y ( 0 + y 2 ⋅ 2 x ) ( 1 + x 2 y 2 ) 2 f x x = − y 3 ⋅ 2 x ( 1 + x 2 y 2 ) 2 f x x = − 2 x y 3 ( 1 + x 2 y 2 ) 2 To find f y x we differentiate the function first with respect to y and then again with respect to x

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11, Problem 30SP

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To Calculate: fxx&fyx for f(x,y)=tan−1xy .

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Chapter 11, Problem 29SP

Chapter 11, Problem 31SP

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A factorization A = PDP 1 is not unique. For A= 7 2 -4 1 1 1 5 0 2 1 one factorization is P = D= and P-1 30 = Use this information with D₁ = to find a matrix P₁ such that - -1 -2 0 3 1 - - 1 05 A-P,D,P P1 (Type an integer or simplified fraction for each matrix element.)

Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 30 -1 - 1 0 -1 400 0 0 1 A= 3 4 3 0 1 3 040 3 1 3 0 0 4 1 0 0 003 -1 0 -1 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A basis for the corresponding eigenspace is { A. There is one distinct eigenvalue, λ = B. In ascending order, the two distinct eigenvalues are λ₁ ... = and 2 = Bases for the corresponding eigenspaces are { and ( ), respectively. C. In ascending order, the three distinct eigenvalues are λ₁ = = 12/2 = and 3 = Bases for the corresponding eigenspaces are {}, }, and { respectively.

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Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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