1. Which one of the following series converges? [ C ]

(A) n#1(n) .                    (B) n .                    (C) n sin ╱ 、.                    (D) n .

2. The series       (_1)n伞1 A (A is a positive constant) [ B ] n#1                   ^n^n + 1

(A) converges absolutely.             (B) converges conditionally.

(C) diverges.                                 (D) converges or diverges depending on the value of A. ~

3. Suppose that the series       an  converges. Which one of the following series must converge? [ D ]

n#1

~                                      ~                                                    ~                                                              ~

4. Given power series        n2 zn , its convergence set is [ D ]

(A) ╱ _ ← 、.                   (B) ← _ ← 、.                   (C) ╱ _ ← ].                   (D) ← _ ← ].

~

5. Suppose that the power series       an (z _ 3)n  converges conditionally at z = _1.  Which one of the

n#1

following statements is always TRUE? [ C ]

~                                                                                                                         ~

(A)       an 4n  is absolutely convergent.                       (B)       an 4n  is conditionally convergent.

n#1                                                                                                              n#1

~                                                                                                                          ~

(C)       an 5n  is divergent.                                            (D)       an 5n  is convergent.

n#1                                                                                                               n#1

~ n

6. The sum of the series        2n   is [ C ] n#1

(A) 8.                                     (B) 4.                                     (C) 2.                                     (D) 1.

7. Suppose that both o- and u- are nonzero vectors. If o-!u- = |o- × u-|, then the angle between o- and u- is [ A ]

π π 5π 7π

8. Suppose that o- and u- are nonzero vectors and o- is not parallel to u-.  Which one of the following statements is NOT TRUE? [ D ]

(A) (o- _ u-) . (o- × u-) = 0 .

(C) (o- × u-) × (u- × o-) = .

(B) (o- × u-) . (u- × o-) ← 0 .

(D) (o- _ u-) × (o- × u-) = .

z + 1 y + 1 3 + 1

(A) The Line L lies in the plane Π –

(B) L is parallel to Π, but does not lie in Π –

(C) L is perpendicular to Π –

(D) L and Π are intersecting but not perpendicular.

10. Find the area of the part of the plane 2z + 3y + 3 = 6 that lies in the first octant? [ A ]

(A) 3^14.                                 (B)6^14.                                 (C) 6.                                 (D) 9.

11. Which one of the following is the equation of the plane that contains the line z = 3u← y = 1 + u← 3 = 2u and parallel to the intersection of the planes y + 3 = _1 and 2z _ y + 3 = 0? [ A ]

(A) 3z _ 5y _ 23 + 5 = 0.

(C) 3z _ 5y _ 23 _ 5 = 0.

(B) 3z + 5y _ 23 _ 5 = 0.

(D) 3z + 5y _ 23 + 5 = 0.

z2 y2

(A)                                           (B)                                           (C)                                           (D)

13. Which of the following limits does not exist? [ D ]

cos(zy) _ 1 ^1 + zy _ 1

^z2 + y2 z2 + y2 .

14. Let f be a differentiable function with f (1← 1) = 2 and fα (1 ← 1) = 6← fg (1 ← 1) = 4.  Use the linear approximation of f (z← y) to approximate f (0 – 98 ← 1 – 01). [ B ]

(A) 1.84.                               (B) 1.92.                               (C) 2.08.                               (D) 2.16.

zy

(A) z _ 4y + 3 = 0.                             (B) z _ 4y _ 3 + 8 = 0.

(C) z + 4y _ 3 _ 8 = 0.                      (D) z + 4y + 3 _ 16 = 0.

16. If f (z← y) = z3 _ 4z2 + 2zy _ y2 , then which one of the following statements is TRUE? [ C ] (A) f (2← 2) is a local minimum value.

(B) f (0← 0) is a local minimum value.

(C) The point (2 ← 2) is a stationary point but f (2← 2) is not a local extreme value.

(D) The point (0 ← 0) is a stationary point but f (0← 0) is not a local extreme value.

17. If R = {(z← y) : 1 ≤ z ≤ 2 ← 0 ≤ y ≤ 2}, then       (z + y)dǎ= [ B ]

R

(A) 4.                                        (B) 5.                                        (C) 6.                                        (D) 8.

3      g

18. (y2 _ z2 )dzdy= [ A ] 1      ó

(A) .                                   (B) _ .                                   (C) 10.                                   (D) _10.

1       ^g

19. Change the order of integration of the iterated integral I =      [        f (z← y)dz]dy . [ D ]

1       ^α 1       1 ó       ó

(A) I =       [          f (z← y)dy]dz.                                (B) I =       [        f (z← y)dy]dz.

ó       ó                                                                                                          ó       ^α

1       α2                                                                                                                                                    1       1

(C) I =      [       f (z← y)dy]dz.                              (D) I =      [      f (z← y)dy]dz. ó       ó                                                                                                  ó       α2

20. Find the surface area of the part of the paraboloid 3 = z2 + y2  that is below the plane 3 = 1.

[ C ]

(5^5 _ 1)π (5^5 _ 1)π

(A) 3π .                              (B) 6π .                              (C)           6          .                              (D)          12         .

II. Calculations and comprehensive problems.

21.

(1) Let f (z← y) = ln ^z2 + y2 . Find fα (z← y), fg (z← y) and fαg (1 ← 1).

(2) Let 3 = 3 (z← y) be an implicit function defined by the equation 32 y _ z33  _ 1 = 0.  Find |)ó ,1 ,1〉 and |)ó ,1 ,1〉.

Solution.

(1) fα (z← y) = z2 + y2 ← fg (z← y) = z2 + y2 ← fαg (z← y) = _ (z2 + y2 )2             fαg (1 ← 1) = _ 2 .

(2) Let F (z← y← 3) = 32 y _ z33 _ 1←    Fα = _33 ←    Fg  = 32 ←    F之 = 23y _ 332 z.

= _F(F)α之 = _ = ← |)ó ,1 ,1〉  = –

= _F(F)g之 = _ = _ ← |)ó ,1 ,1〉  = _ –

22. The temperature at (z← y← 3) of a solid sphere centered at the origin is T (z← y← 3) = e_ )α2 伞g2 伞之2〉.

(1) Find the directional derivative of T at (1 ← 1 ← _1) in the direction of u- = (1← 2 ← 2) .

(2) In which direction at (1 ← 1 ← _1), the temperature increases most rapidly? Find the maximum rate of change.

Solution.

(1) Tα = e_ )α2 伞g2 伞之2〉(_2z)← Tg  = e_ )α2 伞g2 伞之2〉(_2y)← T之 = e_ )α2 伞g2 伞之2〉(_23) 9T (1← 1 ← _1) = e_3 (_2 ← _2 ← 2) ← o- = = ( ← ← )

DU→T (1← 1 ← _1) = 9T (1← 1 ← _1) ! o- = _

(2) In 9T (1← 1 ← _1) = e_3 (_2 ← _2 ← 2), the temperature increases most rapidly.

The maximum rate of change is |9T (1← 1 ← _1)| = 2e_3 ^3.

23. Find the global maximum value and the global minimum value of f (z← y) = 2z2 + y2 + zy on the closed bounded set s = {(z← y) : z2 + y2  ≤ 1}.

Solution. Method I

From , we get the stationary point (0 ← 0).

On the boundary of s, let z = cos u← y = sin u← u e [0 ← 2π]. Then

F (u) := f (z(u)← y(u)) = 2 cos2 u + sin2 u + cos u sin u = cos 2u + sin 2u + –

Solving F\ (u) = _ sin 2u + cos 2u = 0 < tan 2u = 1 < 2u = 4 ← 2u = 4 ← 2u = 4 or 2u = 4 , we get

π 5π 9π 13π

Comparing with the values of function f (z← y) = 2z2 + y2 + zy on the following points

π π 9π 9π 3 +^2

8        8                 8         8               2 ←

5π 5π 13π 13π 3 _ ^2

f (z( 8  )← y( 8  )) = f (z(   8   )← y(   8   )) =       2      ← f (z(0)← y(0)) = f (z(2π)← y(2π)) = 2←

3 +^2

2

Method II

From , we get the stationary point (0 ← 0).

On the boundary of s:  (z← y)← z2 + y2 = 1.

Let F (z← y← A) = 2z2 + y2 + zy + A(z2 + y2 _ 1). Then

,．Fα = 4z + y + 2Az = 0       (1)

．

．

．

．‘Fλ = z2 + y2 _ 1 = 0          (3)

(1)y _ (2)z, we have 4zy + y2 _ (2zy + z2 ) = 0 ÷ (z + y)2 _ 2z2 = 0, then

,．z + y = ^2z

．

． or

．

,．y = (^2 _ 1)z

．

÷ ．or

．

Substituting y = (^2 _ 1)z into (3), we have

z2 + (^2 _ 1)2 z2 = 1     ÷ z2 =                  ÷ z = ±

which corresponds to

(z1 ← y1 ) = ← 、← (z2 ← y2 ) = ╱ _ ← _ ← 、←

Substituting y = _(^2 + 1)z into (3), we have

z2 + (^2 + 1)2 z2 = 1     ÷ z2 =                  ÷ z = ±

which corresponds to

(z3 ← y3 ) = ← 、← (z4 ← y4 ) = ╱ _ ← ← 、←

Comparing the following function values

3 +^2 3 _ ^2

f (0← 0) = 0← f (z1 ← y1 ) = f (z2 ← y2 ) =       2      ← f (z3 ← y3 ) = f (z4 ← y4 ) =       2      ←

3 +^2

2

24. Find the volume of the solid in the first octant that is inside the sphere z2 + y2 + 32 = 1 and outside the cylinder z2 + y2 = z.

Solution.

The trace of the sphere on zy _plane is

that is, The trace of the cylinder on zy _plane is , that is,．

3 = ^1 _ z2 _ y2 , we have

1

4

.

0

s = {(z← y) : 0 ≤ z ≤ 1← 0 ≤ y ≤ (1/4 _ (z _ )2 } = {(t← θ) : 0 ≤ t ≤ cos θ← 0 ≤ θ ≤ π/2} –