Final 2005 Fall

Kuwait University Dept: Math & Comp Sci

Math 211 Final Exam

Date: Jan. 23, 2006 Duration: 120 minutes

Calculators and mobile telephones are NOT allowed. Each question is 4 points. 1. For each statement, state whether it is TRUE or FALSE. If it is true, give a proof; if it is false, provide a counter-example. ∞ X (a) If an converges, then lim an = 0. n→∞

n=1

(b) If lim an = 0, then n→∞

∞ X

an converges.

n=1

2. Determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent. ∞ ∞ X X n (ln n)2 n (a) (−1) (b) (−1)n sin(1/n) n n=1 n=1 3. Find the points on the ellipsoid x2 + 2y 2 + 3z 2 = 13 where the tangent plane is parallel to the plane 2x + 4y − 3z = 24. 4. Find the maximum and minimum value of the function f (x, y) = 2x2 − y 2 + y + 3 over the  half-disk D = (x, y) : x2 + y 2 ≤ 1, y ≥ 0 . 5. Reverse the order of integration and evaluate

Z

9 0

Z

3 √

sin(x3 ) dx dy. y

6. Let R be the region in the first octant which is bounded by the coordinate planes, the cylinder y = 4 − x2 and the plane z = 6. Find the average value of the function f (x, y, z) = x over R. p 7. Let D be the region bounded by the cone z = 2x2 + 2y 2 and the paraboloid z = 8 − x2 − y 2 . Z Z Z Use cylindrical coordinates to express f (x, y, z) dV as a triple iterated integral. D

8. Evaluate the line integral

Z

yz dx + xz dy + xy dz C

if C is the line segment from A(−1, 1, 0) to B(2, 1, 4). −y x i + j be a vector field over R2 \ {(0, 0)}, and consider the two paths x2 + y 2 x2 + y 2 from A(−1, 0) to B(1, 0) given by

9. Let F(x, y) =

C1 : x = − cos t,

y = sin t,

0 ≤ t ≤ π;

C2 : x = cos t,

y = sin t,

π ≤ t ≤ 2π.

R R (a) Evaluate C1 F · dr and C2 F · dr. (b) Is F conservative? Explain. 10. Use Green’s theorem to evaluate the line integral simple closed curve x = cos t, y = sin t, 0 ≤ t ≤ 2π.

I

2

(y + ex ) dx + (2x − cos(y 3 )) dy if C is the C

Dept: Math & Comp Sci

Incomplete Exam

Duration: 120 minutes

Calculators and mobile telephones are NOT allowed. Each question is 4 points. 1. (a) State the definition of an absolutely convergent series. P P (b) Prove that if an is absolutely convergent, then an is convergent. P (c) Give an example of a convergent series cn which is not absolutely convergent. 2. Determine whether each of the following series converges or diverges. ∞ ∞ 2 2 X X n n n − cos n (a) (−1) (b) (−1)n . 1/n (n + 1)(n2 + 1) n n=1 n=1 ∞ X (xy)n . 3. Describe and sketch the domain of the function f (x, y) = n n=1

4. Show that the function f (x, y) =



xy x2 +xy+y 2

if (x, y) 6= (0, 0)

0

if (x, y) = (0, 0)

is not continuous at (0, 0).

5. Express the area of the region bounded by y = x2 and y = x + 2 using iterated double integrals (a) in the order dy dx (b) in the order dx dy.

6. Let Q be the solid in the first octant which is bounded by the coordinate planes and the plane x + y + z = 3. Find the average of f (x, y, z) = z over Q.

7. Express

Z Z Z

f (x, y, z) dV as an iterated triple integral if D is the solid that lies outside the D

sphere x2 + y 2 + z 2 = 6 and inside the sphere (x − 1)2 + (y − 1)2 + (z − 2)2 = 1.

8. Evaluate

Z

(2xy − x2 ) dx + (x + y 2 ) dy if C is the curve y = x2 from (0, 0) to (4, 2). C

9. Let F(x, y) = (2x sinh x2 + yxy−1 )i + (xy ln x − sec2 y)j be a vector field over the half-plane D = {(x, y) : x > 0}. R (a) Show thatZ the line integral C F · dr is independent of path. (b) Evaluate

F · dr if C is a smooth path from A(1, 1) to B(2, 3).

C

10. Use Green’s theorem to find the area bounded by curve x = 3 − 2 cos t, y = 5 + 4 sin t, 0 ≤ t ≤ 2π.