Explanation Given information: The performance of the subject on a certain task is given by f ( x , y ) = 10 + x y e 1 − x 2 − y 2 where x and y are units of stimuli A and B. Also, given function works for combination ( x , y ) with x ≥ 0 , y ≥ 0 , and x + y ≤ 1 Calculation: To find out the average response, the average value for a function of two variables is given by f a v g ( x , y ) = 1 A ∬ f ( x , y ) d x d y Here, ‘ A’ the area of the region covered by x ≥ 0 , y ≥ 0 , and x + y ≤ 1 The common area covered by the three inequalities is in the shape of a triangle with points ( 0 , 0 ) , ( 1 , 0 ) and ( 0 , 1 ) . Area = A = 0.5 Both the value of x and y range from 0 to 1. Substituting these values in the formula for average value defined above, f a v g ( x , y ) = 1 0.5 ∬ ( 10 + x y e 1 − x 2 − y 2 ) d x d y = 2 ∫ 0 1 ∫ 0 1 ( 10 + x y e 1 − x 2 − y 2 ) d x d y ------- (1) Since it’s a double integral, first integrate the function with respect to “ dx ” by taking ‘ y’ as a constant. f a v g ( x , y ) = 2 ∫ 0 1 [ ∫ 0 1 ( 10 + x y e 1 − x 2 − y 2 ) d x ] d y Taking the function present in the square-brackets aside to perform integration· ∫ 0 1 ( 10 + x y e 1 − x 2 − y 2 ) d x Going by “ u -substitution”, 1 − x 2 − y 2 = u d u d x = − 2 x d u = − 2 x d x x d x = − d u u Substitute the above in place of “ xdx ” and integrate, ∫ 0 1 ( 10 + x y e 1 − x 2 − y 2 ) d x = 10 [ x ] 0 1 + y ∫ 0 1 x e 1 − x 2 − y 2 d x = 10 − y 2 ∫ 0 1 e u d u = 10 − y 2 [ e u ] 0 1 = 10 − y 2 [ e 1 − x 2 − y 2</

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

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

Publisher: Kendall Hunt Publishing

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Chapter 12, Problem 86SP

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To calculate: The subject’s average response to stimuli, rounded off to the nearest hundredth·

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Chapter 12, Problem 85SP

Chapter 12, Problem 87SP

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Chapter 12 Solutions

Student's Solution and Survival Manual for Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

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The subject’s average response to stimuli, rounded off to the nearest hundredth·

Solution Summary: The author calculates the subject's average response to stimuli, rounded off to the nearest hundredth.

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Chapter 12 Solutions