Concordia University
Final Examination - EMAT 233 - All sections

Date: December 2003
Time Allowed: 3 hours
Instructors: G. Pusztai, P. Gauthier
Course Examiner: C. David
Directions: Answer all questions. NO CALCULATORS.

MARKS
(12)

1. Let C be the curve C : r(t) = (t, t2 , t3 ) in R3 .
(a) Find the tangential component of acceleration at t = 1.
(b) Find the equation of the tangent line at the point P = (2, 4, 8).

(8)

2. If f (x, y) = xy tan

y
, show that x x

(16)

∂f
∂f
+y
= 2f.
∂x
∂y

3. (a) Find the equation of the tangent plane to the surface x2 − 2y 2 + 3z 2 = 5 at the point P = (2, 1, 1).
(b) Find the directional derivative of the function f (x, y, z) = xy sin z at the point
P = (1, 2, π/2) and in the direction of the vector v = i + 2j + 2k.

(8)

4. Compute the line integral cos x dx + z dy + y dz
C

where C is any curve starting at (0, −1, −1) and ending at (π, 3, 2).

(16)

5. Consider the line integral y dx + 4x dy
C

where C is the triangle with vertices (0, 0), (2, 0) and (1, 1) oriented counterclockwise.
(a) Evaluate the line integral directly by parametrising the curve C.
(b) Evaluate the line integral using Green’s Theorem.
(8)

6. Evaluate the double integral by first reversing the order of integration
4
y=0

2

3

√

ex dx dy.

x= y

(8)

7. Find the volume of the region between the paraboloids z = x2 +y 2 and z = 8−x2 −y 2 .

(8)

8. Find a unit normal vector n to the surface S parametrised by r(u, v) = (2 cos u, 2 sin u, 3v)
√ √ √ at the point ( 2,