c7sol-1 - Chapter 7 Exercises 7.1 1. f ( x , y) = x 2 3xy y...

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227 Chapter 7 Exercises 7.1 1. ; f (5, 0) = 25 f (5, –2) = 51; f ( x , y ) = x 2 3 xy y 2 f ( a , b ) = a 2 3 ab b 2 2. 2 (, ) 2 2 gxy x y =+ ; g (1, 1) = 3 22 (0, 1) 2; ( , ) 2 g gab a b −= = + 3. g ( x , y , z ) = x y z ; g (2, 3, 4) = –2 g (7, 46, 44) = 7 2 4. 2 (, , ) yz fxyz xe + = ; 2 (1, 1,1) fe f (2, 3, 4) = 4 e 5 5. f ( x , y ) = xy ; f (2 + h , 3) = (2 + h )3 = 6 + 3 h f (2, 3) = 6; f (2 + h , 3) – f (2, 3) = 6 + 3 h – 6 = 3 h 6. f ( x , y ) = xy ; f (2, 3 + k ) = 2(3 + k ) = 6 + 2 k f (2, 3) = 6; f (2, 3 + k ) – f (2, 3) = 6 + 2 k – 6 = 2 k 7. C ( x , y , z ) is the cost of materials for the rectangular box with dimensions x , y , z in feet. C ( x , y , z ) = 6 xy + 10 xz + 10 yz 8. C ( x , y , z ) is the cost of material. C ( x , y , z ) = 3 xy + 5 xz + 10 yz 9. f ( x , y ) = 20 x 1/3 y 2/3 ; f (8, 1) = 40 f (1, 27) = 180; f (8, 27) = 360 f (8 k ,27 k ) = 20(8 k ) (27 k ) = 20(8 1/3 )( k 1/3 )(27 )( k 2/ 3 ) = k (20)(8 )(27 ) = kf (8, 27) 10. f ( x , y ) = 10 x 2/5 y 3/5 f (3 a ,3 b ) = 10(3 a ) b ) = 3 f ( a , b ) = 10(3 )( a )(3 )( b ) = 3(10)( a )( b 3/5 ) 11. P ( A , t ) = Ae .05 t 50.16 $50 invested at 5% continuously compounded interest will yield $100 in 13.8 years. P (100, 13.8) = 100 e .05(13.8) = 100 e .69 12. C ( x , y ) is the cost of utilizing x units of labor and y units of capital. C ( x , y ) = 100 x + 200 y 13. a. v = 200,000, x = 5000, r = 2.5 T = r (.4 v x ) 100 = 2.5(.4(200, 000) 5000) 100 T = $1875 b. If v = 200,000, x = 5000, r = 3: T = r (.4 v x ) 100 = 3(.4(200,000) 5000) 100 T = $2250 The tax due also increases by 20% 1 5 since 1875 + 1 5 (1875) = $2250 . 14. a. v = 100,000, x = 5000, r = 2.2 T = r (.4 v x ) 100 = 2.2(.4(100, 000) 5000) 100 T = $770 b. If v = 120,000, x = 5000, r = 2.2: T = r (.4 v x ) 100 = 2.2(.4(120, 000) 5000) 100 T = $946 20% of $770 is $154, so tax due does not increase by 20%. 15. C = 2 x - y , so y = 2 x C 16. C =− x 2 + y , so y = x 2 + C
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Chapter 7: Functions of Several Variables ISM: Calculus & Its Applications, 11e 228 17. C = x y , y = x C But 0 = 0 – C C = 0, so y = x . 18. C = xy , y = C x But 42 1 2 C C =⇒= , thus 2 y x = . 19. y = 3 x – 4, y – 3 x = –4: so y – 3 x = C , some constant f ( x , y ) = y – 3 x . 20. 2 2 2 , 2 yy x x == Thus, yx 2 = C , some constant f ( x , y ) = x 2 y . 21. They correspond to the points having the same altitude above sea level. 22. C ( x , y ) = 100 x + 200 y is the cost of using x units of labor and y units of capital. If C ( x , y ) = 600, then 100 x + 200 y = 600; y = 3 1 2 x . If C ( x , y ) = 800, then y = 4 1 2 x . If C ( x , y ) = 1000, then y = 5 1 2 x . Points on the same level curve correspond to production amounts that have the same total cost. x y 10 1000 800 600 8 6 3 4 5 23, 24, 25, 26. 25) is matched with (c). Imagine slicing “near the top” of the “4 humps”; we get a cross-section of 4 circular-like figures. As we move further down these figures become larger. Similarly 23) is matched with (d), 24) is matched with (b), 26) is matched with (a). Exercises 7.2 1. f ( x , y ) = 5 xy ; f x = 5(1) y = 5 y ; f y = x = 5 x 2. 22 (, ) ; 2; 2 ff fxy x y x y xy ∂∂ = −== 3. f ( x , y ) = 2 x 2 e y ; f x = 2(2 x ) e y = 4 xe y f y = 2 x 2 e y (1) = 2 x 2 e y 4. ; () ( 1 ) x yx y f x y f x y xe xe y e x ==+ 2 0 ) x y f x y x ex e x e y =+= 5. 11 2 1 ; x yf x y x y y x −− + =− ; 2 1 fx y = −+
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ISM: Calculus & Its Applications, 11e Chapter 7: Functions of Several Variables 229 6. 2 11 (, ) ; () f fxy xy x x y == +∂ + 2 1 f y x y = ∂+ 7. f ( x , y ) = (2 x y + 5) 2 ; f x = 2(2 x y + 5)(2) = 4(2 x y + 5) f y = 2(2 x y + 5)(–1) = –2(2 x y + 5) 8.
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c7sol-1 - Chapter 7 Exercises 7.1 1. f ( x , y) = x 2 3xy y...

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