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MATH 223 FINAL EXAM REVIEW PACKET ANSWERS

(Fall 2012) 1. (a) increasing (b) decreasing 2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis with radius 5. (b) 3x = , 3x = − . These are vertical planes parallel to the yz-plane. (c) 2 2 2z x y= + . This is a cone (one opening up and one opening down) centered on the z-axis. 3. There are many possible answers. (a) 0x = produces the curve 23y z= − .

(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − − .

(c) 2

x π

= produces the curves 3z = and 3z = − .

4. (a) (b) (i) 1 (ii) Increase (iii) Decrease 5. (a) Paraboloids centered on the x-axis, opening up in the positive x direction. 2 2x y z c= + + (b) Spheres centered at the origin with radius 1 ln c− for 0 c e< ≤ . 2 2 2 1 lnx y z c+ + = − 6. (a) 6 am 11:30 am (b) Temperature as a function of time at a depth of 20 cm. (c) Temperature as a function of depth at noon.

7. ( , ) 2 3 2z f x y x y= = − −

8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI (e) I, V

9. (a) 12

4 12 5

z x y= − + (b) There are many possible answers. 12

4 5

i j k+ − 

 

(c) 3 569

2

10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f) i, ix 11. There are many possible answers.

(a) ( )5 4 3 26

i j k− + 

 

or ( )5 4 3 26

i j k− − + 

 

(b) 2 3i j− +  

(c) 4

cos 442

θ = , 1.38θ ≈ radians (d) ( )4 4 3 26

i j k− + 

 

(e) 4 11 17i j k− − − 

 

12. (a) 3 5

a = − (b) 1 3

a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)

1 2 , 2 , 3 3x t y t z t= + = − − = + 13. 6 39i



or 6 39i− 

14. (a) ( )

2

23 2 2

3 2 3 1

z x y x x x y x y

∂ = −

∂ + + + (b)

( )4 10 4 3

5 H

H T f

H + +

= −

(c) 2

2 2

1 1z x y y x ∂

= − − ∂ ∂

15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3) 8( 3) 6( 6) 0x y z− + − + − =

16. (a) 2sin(2 ) cos(2 )

5 5 v v

ds dv d α α

α= +

(b) The distance s decreases if the angle α increases and the initial speed v remains constant. (c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.

17. (a) The water is getting shallower. 4

( 1, 2) 17

uh − = −

(b) There are many possible answers. 3i j+  

(c) 72 ft/min

18. (a) ( )

2 2 2

22 2 22

2 2 1 1 11

yz xyz z yz grad i j k

x x xx

  = − + + + + +  +



 

(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = + −  

   

(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k x x x y y e+ + = − + + 

 

(d) 37 3

(e) ( , , ) sin zg x y z xy e c= + + 19. ( , ) 4 3vG a b = − 20. (a) positive (b) negative (c) negative (d) negative (e) positive (f) zero

21. (a) (1, 1 2)

15 8

w u

π ∂

= ∂

and (1, 1 2)

15 4

w v

π ∂

= − ∂

(b) 1

3

t

dw dt e=

= −

22. ( )1, 2 37rz π = 23. ( 1, 3)− − local maximum, (2,1) local minimum, ( 1,1)− and (2, 3)− saddle points 24. (b) 4K < , saddle point, 4K > local minimum, no values of K for local maximum. 25. The minimum distance from the surface to the origin is 6 . This occurs at the points (2, 1,1)− and (2, 1, 1)− − .

26. (a) 8r = (b) 4 π

θ = (c) 5 4 π

φ = (d) 10

cos ρ

φ =

27. (a) positive (b) positive (c) negative (d) negative

28. (a) ( )1 sin18 cos 5 cos11 6

− (b) 3 2

(5) 3 π volume of a half sphere

(c) ( )3 22 28 1 9

− change the order (d) 8 1 1

2 3 3 e

π − −   

convert to spherical

(e) 7 3

− (f) 25 2

area of a triangle

29. (a) 2 2 2

2 2 2

2 2 4

2 2

x x y

x x y dzdydx

− − −

− − − +∫ ∫ ∫ 2 2 2 4

0 0

r

r rdzdrd

π θ

−

∫ ∫ ∫

2 4 2 2 0 0 0

sin d d d π π

ρ φ ρ φ θ∫ ∫ ∫

(b) 2 2 2

2 2 2

3 9 20

3 9 2

z y z

z y z dxdydz

− − −

− − − + +∫ ∫ ∫ 2

2

2 3 20

0 0 2

r

r rdxdrd

π θ

−

+∫ ∫ ∫

(c) 2 4 12 16

0 0 0

z dxdydz

−

∫ ∫ ∫ 2 4 12

0 0 0 rdydrd

π θ∫ ∫ ∫

(d) 0 1 3 2 2 6 12

6 0 0

x x y dzdydx

+ − +

−∫ ∫ ∫

30. 2 4 4

0 0 0 (4 )

r k z rdzdrd

π θ

− −∫ ∫ ∫

31. There are many possible answers. (a) 3 cos 2, 1, 3sin 0 2x t y z t t π= + = = − ≤ ≤ (b) 1 2 , 2 3 , 3 x t y t z t t= + = − − = − −∞ < < ∞

(c) ( )3, 2 2 0x t y t t= − = − + − ≤ ≤ (d) 2 cos , 2 sin , 2 0 2x t y t z t π= = = ≤ ≤ 32. (a) 4t = seconds (b) 10 feet per second (c) There are many possible answers. 3, 3( 2 ), 10 2 ( 2 ) 2x y t z t tπ π π π= = − = − − − ≥

33. (a) There are many possible answers. 1 2 , 1 6 , 7+ 0x t y t z t t= + = + = ≥ (b) No 34. (a) iii (b) v (c) vi (d) i (e) ii (f) iv 35.

1 2 3C C C F dr F dr F dr⋅ < ⋅ < ⋅∫ ∫ ∫   

  

36. (a) 21 2

− (b) 10 cos 2− (c) 18π (d) 4

1 16 π

− (e) 1875

2 π

37. (a) 12 2

− (b) 320π (c) 450π (d)

1875 2 π

38. (a) 0 (b) 0 39. 0p = , 500flux π= 40. 22, 000, 000 40, 000, 000 18, 000, 000π π π− = − 41. (a) (i) 0 (ii) 0 (iii) zero k



component (iv) could be a gradient field (b) (i) positive (ii) 0 (iii) positive k



component (iv) could not be a gradient field (c) (i) 0 (ii) positive (iii) zero k



component (iv) could be a gradient field 42. (a) On a sphere of radius 5. (b)

6 5 1S S S F dA F dA F dA⋅ < ⋅ < ⋅∫ ∫ ∫

    

43. 75 12 63π π π− = 44. (a) 10− (b) 29 45. (a) V (b) S (c) S (d) V (e) S (f) V (g) S (h) ND (i) ND 46. (a) false (b) true (c) true (d) true (e) true

47. There are many possible answers.

(a) 10 10 3 2

i j−  

(b) 8 2

− (c) (18.5, 74.5) (d) 10 (e) 6(60 80 50 70) 1560+ + + =