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Simon Fraser University
MATH 251 - Summer 2005
Final Exam
Aug 5, 2005, 8:30 โ 11:30 am
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Instructions:
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- This exam contains 15 pages with a total
of 11 questions. Once the exam begins
please check to make sure your exam
booklet is complete.
- Only complete well-organized solution
will receive full credit
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the space on the back of the previous
page and clearly indicate where the
solution continues.
- Only scientific calculators are allowed.
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usual writing instruments, this booklet
and a scientific calculator, shall be
within reach of a student during the
examination.
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communicating with, or deliberately
exposing written papers to the view of
other examinees is forbidden.
Question Marks
Total /
- Suppose that a moving point has initial position vector r (0) = 2 i
K
K
, and velocity
vector. Find its position vector
2
v t ( ) = (1 โ 2 ) t i + (3 t โ1) j
K
K
K
r t ( )
K
and acceleration
vector a t ( ). [6 marks]
K
b) Find the equation of the line that is tangent to the curve C and that is
parallel to the line: x = โ + t 1, y = t + 2, z = 2 t + 3. [6 marks]
a) Determine all values of a such that the vector
2
v =< a , โ2 , a โ 1 >
K
lies in the
plane tangent to the surface / at the point. [6 marks]
x
z = e y (0,1,1)
- Find the limit
3
2
( , ) (0,0)
lim
x y
x
2
x y
โ
, if it exists. [6 marks]
- Find the linear approximation of f ( , x y ) = 4 x + y + 1 at the point. Use
this to estimate
4 1.2 + 3.9+ 1. [7 marks]
a) Evaluate
1 1
0
sin
y
x
dxdy
x
by first reversing the order of integration.
[7 marks]
b) Find the volume of the solid region that lies below the surface
2 2
xy
z
x y
and above the plane region R. R is bounded by the graphs of xy = 1 , xy = 4 ,
x = 1 and x = 4. (Hint: Let u = x , v = xy .) [8 marks]
- Find the moment of inertia
2 2
z
T
I = x + y ฮด x y z dV
of the solid T
with ฮด ( , x y z , ) = z , where the solid T lies between the spheres
2 2 2
x + y + z = 1
and in the first octant. [8 marks]
2 2 2
x + y + z = 9
a) Given
2
f ( , x y z , ) = tan x + z ln y , find div ( โ f )and curl ( โ f ). [5 marks]
b) Determine if the vector field
2 2
xy xy
F x y = x + ye i + y + xe j
K
K K
is
conservative. [4 marks]
