LAST (family) NAME: FIRST (given) NAME: ID # : MATHEMATICS 2Q04 McMaster University Final Examination Day Class Duration of Examination: 3 hours THIS EXAMINATION PAPER INCLUDES 24 PAGES AND 15 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR. Instructions: • Indicate your answers clearly in the spaces provided. In the full answer questions in Part I (questions 1 - 5) show all your work to receive full credit. For the multiple choice questions in Part II (questions 6 - 15) be sure to circle the correct letters on page 12 to receive full credit. • No books, notes, or “cheat sheets” allowed. • The only calculator permitted is the McMaster Standard Calculator, the Casio fx 991. • The total number of points is 100. • There is a formula sheet included with the exam on page 24. • Pages 23 of the test is for scratch work or overflow. If you continue the solution to a question on this page, you must indicate this clearly on the page containing the original question. GOOD LUCK!

SAMPLE FINAL EXAM C #

Mark

#

1.

4.

2.

5.

3.

6-15.

Mark

TOTAL Continued. . .

Final Exam / Math 2Q04 NAME:

-2ID #:

Part I: Provide all details and fully justify your answer in order to receive credit. 1. Let D be the region in the first quadrant of the x, y plane (i.e. the region where x, y ≥ 0) where 1 ≤ y ex ≤ 2 and 21 ≤ y e−x ≤ 1. Consider the change of variables ½ x ye = u y e−x = v (a) (4 pts.) Find the region D∗ in the u, v plane that corresponds to D when the change of variable above is used and find expressions for the variables x and y in terms of u and v for a point (x, y) in D.

(b) (2 pts.) Compute the Jacobian

Continued. . .

∂(x, y) corresponding to the change of variables above. ∂(u, v)

Final Exam / Math 2Q04

-3-

NAME:

ID #:

(c) (6 pts.) Use the change of variables above to compute the integral ZZ y 3 dx dy. D

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

2. (a) (6 pts.) Let S be the part of the hemisphere (x + 1)2 + y 2 + z 2 = 4, z ≥ 0 which is inside the cylinder x2 + y 2 = 1. Let C be the curve of intersection of the hemisphere and the cylinder, oriented counterclockwise as viewed from above. Consider the vector field F(x, y, z) = (y) i + (x) j + (y z) k. After finding a suitable parametrization for C, compute the line integral I F · dr C

using the definition of the integral of a vector-field on an oriented curve.

Continued. . .

Final Exam / Math 2Q04 NAME:

-5ID #:

(b) (6 pts.) Compute the line integral in part (a) using Stokes’ theorem applied to the surface S (with an orientation compatible with the orientation of the boundary curve C in part (a) ).

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

3. Consider the wave equation ∂ 2u ∂2u = 4 , ∂t2 ∂x2

0 < x < π, t > 0,

(1)

together with the boundary conditions u(0, t) = u(π, t) = 0,

t > 0,

(2)

0 < x < π.

(3)

and the initial conditions u(x, 0) = 0, ut (x, 0) = 1,

(a) (3 pts.) Using the method of separation of variables, find particular non-trivial solutions of the equations (1) and (2) of the form u(x, t) = X(x) T (t). Show that this leads to the eigenvalue problem X 00 (x)+λ X(x) = 0 with boundary condition X(0) = X(π) = 0 for X(x) together with the differential equation T 00 (y) + 4 λ T (t) = 0 for T (t).

(b) (3 pts.) Find all eigenvalues and eigenfunctions for the eigenvalue problem associated with X(x) in the case where λ > 0. (They are no eigenvalue when λ ≤ 0, but you dont need to show that.)

Continued. . .

Final Exam / Math 2Q04 NAME:

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(c) (3 pts.) For each eigenvalue λ found in part (b), find the non-trivial solutions of the form u(x, t) = X(x) T (t) of (1) and (2) associated with λ and use the superposition principle to find the general solution of (1) and (2).

(d) (3 pts.) Use the result in part (c) to obtain the unique solution of (1) and (2) that also satisfies the boundary conditions (3).

Continued. . .

Final Exam / Math 2Q04

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NAME:

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4. (12 pts.) Let R be the solid region inside the sphere x2 + y 2 + z 2 = 4 and above the plane z = −1. Let S be the boundary surface of R. Consider the vector field F(x, y, z) = x z 2 i + y z 2 j + 2 z (x2 + y 2 ) k. If S is oriented using the outward pointing normal, compute the surface integral ZZ F · n dS S

using the divergence theorem. Warning: Do not waste your time calculating the surface integral directly. No credit will be given for that.

Continued. . .

Final Exam / Math 2Q04 NAME:

Continued. . .

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Final Exam / Math 2Q04 NAME:

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5. Let S be the surface parametrized by the vector function r(u, v) = hv cos u, v sin u, v 2 i, (a) (6 pts.) Compute the surface area of S.

Continued. . .

for 0 ≤ u ≤ 2 π, 0 ≤ v ≤ 2.

Final Exam / Math 2Q04

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(b) (6 pts.) If F(x, y, z) = y 2 i + x2 j + z 2 k, compute the surface integral ZZ F · n dS S

where the unit normal n corresponds to the given parametrization for S (i. e. n has the same direction as ru × rv ).

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

PART II: Multiple choice part. Circle the letter corresponding to the correct answer for each of the questions 6 to 15 in the box below. Indicate your choice very clearly. Ambiguous answers will be marked as wrong. There is only one correct choice for each question and an incorrect answer scores 0 marks.

QUESTION #

V1 Continued. . .

ANSWER:

6.

A

B

C

D

E

7.

A

B

C

D

E

8.

A

B

C

D

E

9.

A

B

C

D

E

10.

A

B

C

D

E

11.

A

B

C

D

E

12.

A

B

C

D

E

13.

A

B

C

D

E

14.

A

B

C

D

E

15.

A

B

C

D

E

Final Exam / Math 2Q04 NAME:

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6. (4 pts.) Compute the directional derivative of the function f (x, y, z) = sin(x + y 2 + z 3 ) at the point P = (−1, 3, −2) in the direction where f decreases the fastest at P . 34 3 √ − 181 √ − 163 √ 96 5 √ 67 − 3

(A) − (B) (C) (D) (E)

(ENTER YOUR ANSWERS ON THE CHART ON PAGE 12.)

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

7. (4 pts.) Let C be the closed curve parametrized by x = cos(t),

y = sin t +

1 cos(2 t), 2

0 ≤ t ≤ 2π

and let D be the region enclosed by the curve C (see the picture below). Use Green’s theorem to compute the area A of the region D. (A) A = π (B) A =

3π 2

(C) A =

2π 3

(D) A =

7π 6

(E) A =

4π 3

0.5

K

1.0

K

0

0.5

K

0.5

K

1.0

K

1.5

Continued. . .

0.5

1.0

Final Exam / Math 2Q04

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NAME:

ID #:

8. (4 pts.) Evaluate the line integral Z I= C

x ds 1 + 2y

where C is the curve parametrized by x = et , y = e2 t , z = 2

(A) I =

e3 − e 3

(B) I =

e2 − 1 4

(C) I = e − 1 (D) I =

e2 − e3 2

(E) I =

e4 − 1 2

Continued. . .

e3 t , 3

0 ≤ t ≤ 2.

Final Exam / Math 2Q04 NAME:

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9. (4 pts.) A particle moves on a curve C parametrized by √ r(t) = h2 + 3 t, 3 − 4 t, 1 + 75 ti, starting at time t = 0. What is the value of the x-coordinate of the particle after it has covered a distance of s = 20 units. (A) x = 3 (B) x = 5 (C) x = 8 (D) x = 10 (E) x = 13

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

10. (4 pts.) A parametric surface S is parametrized by r(u, v) = hu2 − v 2 , u + v, u vi,

−∞ < u, v < ∞.

The equation for the plane tangent to S at the point (−3, 3, 2) is: (A)

x+y+z =2

(B)

x + 10 y − 6 z = 15

(C)

2 x + 3 y − 4 z = −5

(D) 4 x + y + 5 z = 1 (E) −2 x + 5 y + z = 23

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

11. (4 pts.) Suppose that the function f (x, y, z) = x z + ex y + y 2 z 2 is a potential function for the vector field F(x, y, z) (i. e. ∇f = F). Let C be the path parametrized by the vector function ­ ® r(t) = 3 − t, 2 + t, t et−1 , 0 ≤ t ≤ 1. The value of the path integral

Z F · dr C

is which of the following: (A) 2 (B) 5 (C) e3 − 2 (D) 11 (E) 3 e + 2

Continued. . .

Final Exam / Math 2Q04 NAME:

-19ID #:

12. (4 pts.) Let V be the solid region bounded by the surfaces z = 2 − x2 and z = y 2 − 2. The volume of V equals (A) 6 π (B) 2 π (C) 5 π (D) 10 π (E) 8 π

Continued. . .

Final Exam / Math 2Q04

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NAME:

ID #:

13. (4 pts.) If spherical coordinates are used to evaluate the triple integral ZZZ z dV, V

p where V is the solid region bounded below by the cone z = x2 + y 2 and above by the sphere of radius 1 centered at (0, 0, 1), by which of the following iterated integrals is the original integral replaced ? Z

2π

Z

π 4

(A) 0

Z

Z

0 2π

Z

0

Z

0

Z

2π

Z

cos φ

ρ3 sin φ cos φ dρ dφ dθ

0 π

Z

cos(2 φ)

(C) 0

0

Z

2π

Z

0

Z

2π

cos φ 2

ρ3 sin φ cos φ dρ dφ dθ

0 π 4

(E) 0

Z

0

Z

ρ3 sin φ cos φ dρ dφ dθ

0 π 6

(D)

ρ3 sin φ cos φ dρ dφ dθ

0 π 3

(B)

2 cos φ

0

Continued. . .

Z

2 sin φ 0

ρ3 sin φ cos φ dρ dφ dθ

Final Exam / Math 2Q04 NAME:

-21ID #:

14. (4 pts.) Compute the volume V of the solid region in the first octant (x, y, z ≥ 0) bounded by the coordinate planes and the plane 2 x + y + 3 z = 6. (A) V = 3 (B) V = 4 (C) V = 5 (D) V = 6 (E) V = 7

Continued. . .

Final Exam / Math 2Q04 NAME:

-22ID #:

15. (4 pts.) Let C be the intersection of the cylinder x2 + y 2 = 1 with the plane y + z = 2. Then, the curvature of C at the point (0, 1, 1) is: (A) 1 (B) (C) (D)

1 2 √

2 √ 2 2

(E) 2

Continued. . .

Final Exam / Math 2Q04 NAME: SCRATCH

Continued. . .

-23ID #:

Final Exam / Math 2Q04

-24-

NAME:

ID #:

Some formulas you may use: T(t) =

r0 (t) , kr0 (t)k

N(t) =

T0 (t) , kT0 (t)k

v·a r0 (t) · r00 (t) aT = = , v kr0 (t)k

κ(t) =

kT0 (t)k kr0 (t) × r00 (t)k = . kr0 (t)k kr0 (t)k3

kv × ak kr0 (t) × r00 (t)k aN = κv = = v kr0 (t)k 2

projb a =

a·b b kbk2

d [u(t) r(t)] = u(t) r0 (t) + u0 (t) r(t), dt d d [r1 · r2 ] = r01 (t) · r2 (t) + r1 (t) · r02 (t), [r1 × r2 ] = r01 (t) × r2 (t) + r1 (t) × r02 (t), dt dt d d (cos t) = − sin t, (sin t) = cos t. dt dt 1 1 1 1 cos2 t = + cos(2t), sin2 t = − cos(2t). 2 2 2 2 2 cos α cos β = cos(α − β) + cos(α + β), 2 sin α sin β = cos(α − β) − cos(α + β), et + e−t et − e−t 2 sin α cos β = sin(α + β) + sin(α − β) cosh t = sinh t = 2 2 p ρ = x2 + y 2 + z 2 , x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ p r = x2 + y 2 = ρ sin φ I ZZ ∂Q ∂P P dx + Q dy = − dA ∂y C D ∂x I ZZ F · dr = (∇ × F) · n dS C

S

ZZ

ZZZ

F · n dS = S

Z L ³ nπx ´ 2 an = f (x) cos dx, 0 < x < L. L 0 L Z L ³ nπx ´ 2 bn = f (x) sin dx, 0 < x < L. L 0 L

³ nπx ´ a0 X f (x) = + an cos , 2 L n=1 ∞

f (x) =

∞ X n=1

*** END ***

bn sin

³ nπx ´ L

∇ · F dV T

,