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 point P (2, 0, 0).
Note that r(0) = 2, 0, 0 and so 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 + 2y
= 2 1, x, y
div F =
(x y 2 ) +
(2z + 1) + (x2 )
x
y
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 parametrise
the circle C. (Note that line integrals with
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
by the formula
volume (S) =
1 dV
S
If we switch to spher
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
=
+
dt
x dt
y dt
= 2x( sin t) + 2y(cos t)
= 2 cos t sin t
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
area(R)
f (x, y) dA
R
We could compute the area of R us
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 continuous at (0, 0) and so
lim
(x,y)(0,0)
2(0)2 + (0) + 3
3
2
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 Chemistry of Advanced
Materials
SF Medicinal Chemistry
Tri
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 have
o
lim t ln t =
+
t0
lim
+
ln t
t0
1
t
= lim
+
t0
1
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: First note that since f (x, y) is a polynomial in two varia