CWA CH 9 FINAL_97_18 - Chapter 9 MULTIVARIABLE CALCULUS 9.1...

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Chapter 9 MULTIVARIABLE CALCULUS 9.1 Functions of Several Variables 1. f ( x;y )=2 x ¡ 3 y +5 (a) f (2 ; ¡ 1) = 2(2) ¡ 3( ¡ 1)+5 = 12 (b) f ( ¡ 4 ; 1) = 2( ¡ 4) ¡ 3(1) + 5 = ¡ 6 (c) f ( ¡ 2 ; ¡ 3) = 2( ¡ 2) ¡ 3( ¡ 3)+5 = 10 (d) f (0 ; 8) = 2(0) ¡ 3(8) + 5 = ¡ 19 2. g ( )= x 2 ¡ 2 xy + y 3 (a) g ( ¡ 2 ; 4) = ( ¡ 2) 2 ¡ 2( ¡ 2)(4) + (4) 3 =84 (b) g ( ¡ 1 ; ¡ 2) = ( ¡ 1) 2 ¡ 2( ¡ 1)( ¡ 2)+( ¡ 2) 3 = ¡ 11 (c) g ( ¡ 2 ; 3) = ( ¡ 2) 2 ¡ 2( ¡ 2)(3) + (3) 3 =43 (d) g (5 ; 1) = (5) 2 ¡ 2(5)(1) + (1) 3 =16 3. h ( p x 2 +2 y 2 (a) h (5 ; 3) = p 25 + 2(9) = p 43 (b) h (2 ; 4) = p 4+32=6 (c) h ( ¡ 1 ; ¡ 3) = p 1+18= p 19 (d) h ( ¡ 3 ; ¡ 1) = p 9+2= p 11 4. f ( p 9 x y log x (a) f (10 ; 2) = p 9(10) +5(2) log10 = p 100 1 =10 (b) f (100 ; 1) = p 9(100) +5(1) log100 = p 905 2 (c) f (1000 ; 0) = p 9(1000) +5(0) log1000 = p 9000 3 p 10 (d) f μ 1 10 ; 5 = q 9 ¡ 1 10 ¢ +5(5) log 1 10 = p 25 : 9 ¡ 1 = ¡ p 25 : 9 5. x + y + z =9 If x =0 and y ;z : If x and z ;y : If y and z ;x : 6. x + y + z =15 To …nd x -intercept, let y : x +0+0=15 x To …nd y -intercept, let x : y To …nd z -intercept, let x : z Sketch the portion of the plane in the …rst octant that contains these intercepts. 580
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Section 9.1 Functions of Several Variables 581 7. 2 x +3 y +4 z =12 If x =0 and y ;z =3 : If x and z ;y =4 : If y and z ;x =6 : 8. 4 x +2 y z =24 x -intercept: y 4 x x y -intercept: x 2 y y z -intercept: x 3 z z =8 Sketch the portion of the plane in the …rst octant that contains these intercepts. 9. x + y If x : If y : There is no z -intercept. 10. y + z =5 x -intercept: y 0=5 Impossible, so no x -intercept y -intercept: x y z -intercept: x z Sketch the portion of the plane in the …rst octant that contains these intercepts and its parallel to the x -axis. 11. x The point (5 ; 0 ; 0) is on the graph. There are no y ¡ or z -intercepts. The plane is parallel to the yz -plane. 12. z No x -intercept, no y -intercept Sketch the portion of the plane in the …rst oc- tant that passes through (0 ; 0 ; 4) parallel to the xy -plane.
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582 Chapter 9 MULTIVARIABLE CALCULUS 13. 3 x +2 y + z =24 For z =0 ; 3 x y : Graph the line 3 x y = 24 in the xy -plane. z =2 ; 3 x y =22 : Graph the line 3 x y = 22 in the plane z : z =4 ; 3 x y =20 : Graph the line 3 x y = 20 in the plane z : 14. 3 x + y z =8 z ; 3 x + y : Graph the line 3 x + y in the xy -plane : z ; 3 x + y : Graph the line 3 x + y in the plane z : z ; 3 x + y : Graph the line 3 x + y in the plane z : 15. y 2 ¡ x = ¡ z z ;x = y 2 : Graph x = y 2 in the xy -plane. z = y 2 : Graph x = y 2 in the plane z : z = y 2 +4 : Graph x = y 2 in the plane z : 16. 2 y ¡ x 2 3 = z If z ; we have y = 1 6 x 2 : The level curve is a parabola in the xy -plane with vertex at the origin. If z ; we have y = 1 6 x 2 +1 : The level curve is a parabola in the plane z with vertex at the point (0 ; 1 ; 2) : If z ; we have y = 1 6 x 2 : The level curve is a parabola in the plane z with vertex at the point (0 ; 2 ; 4) : Sketch portions of these curves in the …rst octant. 18. x -intercept: y ;z =0 ( a 6 =0) ax = d x = d a y -intercept: x b 6 by = d y = d b z -intercept: x ;y c 6 cz = d z = d c If d 6 , then for the plane to have a portion in the …rst octant, one of d a ; d b ,or d c must be positive, so at least one of a;b; or c musthavethesamesign as d .
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This note was uploaded on 10/07/2010 for the course CALC 2302 taught by Professor Yuly during the Fall '08 term at University of Texas at Dallas, Richardson.

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CWA CH 9 FINAL_97_18 - Chapter 9 MULTIVARIABLE CALCULUS 9.1...

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