MA22S1: TUTORIAL 2 SOLUTIONS
Let C be the smooth space curve with vector equation
r(t) = 2 cos t, 2 sin t, t , 0 t 4
1. Find the unit tangent vector, unit normal vector and binormal vector
to C at the
MA22S1: SOLUTIONS TO TUTORIAL 7
1. Compute the curl and the divergence of the following vector eld,
F(x, y, z) = x y 2 , 2z + 1, x2
Solution:
curl F =
F
i
=
j
k
x
y
z
x y 2 2z + 1 x2
=
0 2, 0 2x, 0 +
MA22S1: SOLUTIONS TO TUTORIAL 6
1. Compute the line integrals
(a)
C
2xy ds
(b)
C
y dx x dy
where C is the circle with centre (0, 0) and radius 2.
Solution: To compute these integrals we need to rst pa
MA22S1: SOLUTIONS TO TUTORIAL 9
1. Find the volume of the region bounded by the sphere
x2 + y 2 + z 2 = 9
and the cone
z=
x2 + y 2
Solution: Let S be the region in question. The volume of S is given
b
MA22S1: SOLUTIONS TO TUTORIAL 4
1. Using the Chain Rule nd
dw
.
dt
(i) w = x2 + y 2 where x = cos t and y = sin t.
(ii) w = z sin (xy) where x = t, y = ln t and z = et1 .
Solution:
(a)
w dx w dy
dw
=
MA22S1: SOLUTIONS TO TUTORIAL 8
1. Find the average value of the function
f (x, y) = x2 y
on the rectangular region
R = [1, 3] [4, 4]
Solution: The average value of f on R is given by
average(f ) =
1
MA22S1: TUTORIAL 3 SOLUTIONS
1. Find the following limit:
lim
(x,y)(0,0)
2x2 + y + 3
x2 y 2 + 2
Solution: Notice that this is a rational function which is dened at
(0, 0). Thus the function is continu
UNIVERSITY OF DUBLIN
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Natural Science
SF Human Genetics
SF Chemistry with Molecular Modelling
SF Physics and Che
MA22S1: TUTORIAL 1 SOLUTIONS
1. Calculate the following limit:
lim cos t, sin t, t ln t
t0+
Solution: We have limt0+ cos t = cos 0 = 1 and limt0+ sin t =
sin 0 = 0. Note that using LHpitals Rule we ha
MA22S1: SOLUTIONS TO TUTORIAL 5
1. Find all critical points of the function f (x, y) = x3 + 3xy + y 3 and
classify each critical point as a local maximum, local minimum or
saddle point.
Solution: Firs