MA1104 Tutorial 3 (Week 5) Practice Problems
Question 1. Determine whether the following limit exists. Justify your answer.
(i)
xy 2
.
(x,y)(2,0) (x 2)2 + y 4 + 4y 2
(ii)
2x2 y 2 sin x
.
(x,y)(0,0) x4 + y 4
lim
lim
Answer.
(i) Along the path y = 0, the li
MA1104 Tutorial 2 (Week 4) Practice Problems
Question. A tetrahedron is a solid with four vertices P , Q, R and S, and four triangular faces as shown
in the figure.
Let v1 , v2 , v3 and v4 be vectors with lengths equal to the areas of the faces opposite t
MA1104 Tutorial 4 (Solution)
Q1.
(a)
fx
fy
fxy
fyx
fxx
fyy
12x2 6xy 2 + 2
6x2 y + 3
12xy
12xy
24x 6y 2
6x2 .
=
=
=
=
=
=
(b) Higher order derivatives of f are continuous at points where they are
defined. So by Clairauts Theorem fuvxyvuv = (fvvv )uuxy . No
MA1104 Tutorial 8 (Week 10)
1. Find the volume of the solid bounded above by the sphere x2 +y 2 +z 2 = 9
and below by the paraboloid 8z = x2 + y 2 .
2. Identify the surface whose equation is given by = sin sin .
3. Evaluate the following integral:
Z
Z Z
MA1104 Tutorial 10 (Week 12)
1. Find a vector function r(u, v) (with an appropriate domain D) which
parameterizes the following surfaces:
(a) the upper half of the ellipsoid 9x2 + 4y 2 + 36z 2 = 36.
(b) the part of the paraboloid z = 9x2 y 2 inside the cy
MA1104 Tutorial 11 (Week 13)
1. Compute the divergence and the curl of the following vector field.
(a) F = hxy, yz, y 2 x3 i
(b) F = h xy , yz , xz i
2. Suppose F = hP (x, y, z), Q(x, y, z), R(x, y, z)i and the functions P , Q
and R have continuous second
MA1104 Tutorial 1 (Solution)
Q1. The scalar projection of b onto a is
compa b =
2(3) + (3)(2) + (4)(1)
ba
4
p
=
= .
2
2
2
|a|
14
3 + (2) + 1
Therefore, the vector projection is
proja b = compa b
a
4
1
6
4
2
= (3i 2j + k) = i + j k.
|a|
7
7
7
14
14
Let be
MA1104 Week1
Vectors, Lines & Planes
1. Distance between two points
A very fundamental concept in mathematics is that of distance. Here, we
want to find the formula for distance in terms of the coordinates of the points.
Suppose we have two points (x1 , y
MA1104 Week 5
Double Integral over region on the xy-plane
1. Double Integral over Rectangle
In this section, we shall introduce double integral of a two-variable function
f (x, y) over a rectangle R. We start by reviewing how we arrive at the
definite int
MA1104 Tutorial 5 (Week 7)
1. Use implicit differentiation to find
z
x
and
z
,
y
given that
3exyz 4xz 2 + x cos y = 2.
2. Suppose u = f (x, y) is a differentiable function on R2 and all its secondorder partial derivatives are continuous, and x = r cos , y
MA1104 Week 11
Surface Integral of Vector Field
1. Surface with Orientation
To define surface integral over vector field, we require the surface to be ORIENTABLE.
Definition 1 (Oriented Surface). A surface S is orientable (or two-sided) if
it is possible
MA1104 Tutorial 2 (Solution)
Q1. A direction of the line is h3, 2, 1i (which is a normal vector to the
plane 3x + 2y z = 6) since we assume that the line is perpendicular to the
plane 3x + 2y z = 6.
A point on the line is (2, 3, 4). Therefore, a vector fu
MA1104 Week 9
Line Integrals
1. Line Integral of Scalar Field
Line integrals were invented in the early 19th century to solve problems
involving:
Fluid flow
Forces
Electricity
Magnetism
There are two types of line integrals:
line integrals of scalar
MA1104 Tutorial 3 (Solution)
Q1.
(a) Let P (x, y) denote a point on the circle. Consider the points O(0, 0),
Q(x, 0) and P (x, y). Let t be the angle measured in the anticlockwise direction from the line OQ to the line OP . Then
y
x
, sin t = .
2
2
As t v
MA1104 Week 8
Change of Variables & Jacobian
1. Plane Transformation
A plane transformation T : (u, v) 7 (x, y) from the uv-plane to the xy-plane
is a set of equations
x = x(u, v),
y = y(u, v).
An example of a plane transformation the function that maps p
MA1104 Week 7
Triple Integral in Cylindrical & Spherical
Coordinates
1. Some Interpretations of Triple Coordinates
Just like double integrals, there are many interpretations of triple integral.
So we do ourselves no favour by just thinking of triple integ
MA1104 Tutorial 3 (Week 5)
1. Find a parametrization of
(a) the circle of radius 2, centered at (0, 0) on the xy-plane.
(b) the curve of intersection of the surfaces x+y 2 z 2 = 1 and x+3y 2 +z 2 = 2.
2. Show that
d
(r(t) s(t) = r0 (t) s(t) + r(t) s0 (t).
MA1104 Tutorial 1 (Week 3)
1. Find the scalar projection and vector projection of b = 2i + 3j 4k onto
a = 3i 2j + k. Find the angle between a and b.
2. Let a = 2i j + k. Write b = 3i j + 2k as the sum of two vectors x
and y, where x is parallel to a and y
MA1104 Week 6
Double Integral over Polar Regions &
Triple Integrals
1. Polar Regions
Any point on the xy-plane can be represented by an ordered pair (r, ) where
r is the distance from the origin to the point
is the angle from the positive x-axis to the
MA1104 Tutorial 7 (Week 9)
1. Evaluate the following iterated integral
2
Z
Z
8x2
(x2 + y 2 )3/2 dy dx.
x
0
2. Find the area of the region D (on the xy-plane) bounded the curve given
by the following equation in polar coordinates:
r = 2 2 sin ,
0 2.
3. Fin
MA1104 Week 3
Partial Derivatives, Chain Rule,
Directional Derivatives
1. Partial Derivative
Recall that for a function f of a single variable x, we define the derivative
function as
f (x + h) f (x)
h0
h
f 0 (x) = lim
To extend this to multivariable funct
MA1104 Tutorial 1 (Week 3) Practice Problems
Question. If r = hx, y, zi, a = h2, 2, 2i and b = h4, 4, 6i, show that the vector equation (r a) (r b) = 0
represents a sphere, and find its center and radius.
Answer.
(r a) (r b) =
(hx, y, zi h2, 2, 2i) (hx, y
Online Practice Problems (Week 4)
Question 1:
Find the equation of the tangent plane to the surface
yz = ln(x + z) at the point (0, 0, 1) .
(A) y z x + 1 = 0
(B) z + y = 0
(C) x + y + z = 1
(D) x + y + z = 0
(E) x + z = 0
Ans= (A)
1
Solution. Let F (x, y,
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 2 EXAMINATION 20142015
MA1104 Multivariable Calculus
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. Please write your matriculation/student number only. Do not write your name.
2. This exam
Online Quiz (Week 3)
Question 1:
Use Chain Rule to find
z
s ,
if
z = er cos , r = 10st, =
(A)
(B)
(C)
(D)
(E)
s
cos
er 10t cos +
s2 + t2
s
sin
er 10t cos
s2 + t2
s
sin
er t cos
s2 + t2
s
cos
er cos +
s2 + t2
s
sin
er 10t cos +
s2 + t2
1
s2 + t2.
Ans=
Practice Problems (Week 2)
Question 1:
Identify the quadric surface given by the equation
x2 + 4z 2 2y = 0.
(A) This is an elliptic paraboloid, symmetric about the x-axis.
(B) This is an elliptic paraboloid, symmetric about the y-axis.
(C) This is an elli
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER I EXAMINATION 2009-2010
MA1104
Multivariable Calculus
December 2009 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination. Each student is allowed to bring two p
MA1104
NATIONAL UNIVERSITY OF SINGAPORE
MA1104 MULTIVARIABLE CALCULUS
(SEMESTER 1: AY 2015-2016)
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. Please write your matriculation/student number only. Do not write your name.
2. This examination paper con
MA1104
Student Number:
NATIONAL UNIVERSITY OF SINGAPORE
MA1104 - Multivariable Calculus
(Semester 2 : AY2013/2014)
Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. Write down your matriculation/student number clearly in the space provided at the top o