Sample Answers for Week 4
math 10121
Series and Complex Numbers
Easy Questions
ei
i 3
1.
(a)
ei = 1
(b)
ei3/2 = i
i
(c)
(e)
e
i4
=1
i/2
e
(f)
2ei/3 = 1 + i 3
(a)
1 = ei0
(b)
i = ei3/2
(c)
i = ei/2
(d)
/4
1+i=
(f)
1i=
(g)
1 + i =
(h)
1 i =
(i)
1 + i 3 = 2e

Problem Sheet for Week 2
math 10121
Numbers and Functions
Reminder :
attempt all easy questions without delay and discuss any diculties in your supervision class
attempt the starred standard questions (see the math10121 web page) and hand in your workin

Problem Sheet for Week 3
math 10121
Limits and Derivatives
Reminder : be sure you can answer all easy questions; hand in the starred standard questions (see the
math10121 web page) for your supervision class, and have a go at the remaining questions; the

Problem Sheet for Week 4
math 10121
Series and Complex Numbers
Easy Questions
1. Using Eulers formula ei = cos + i sin write the following complex numbers in the standard form
(real part + i imaginary part). Sketch all of the points on a single diagram of

MATH10121 A2
A2.1
sketching some simple functions
Functions
We can say that
A2.2
Revision
1 t2 2t
= 2 (t + 1)2
x2
a function maps any
element in its domain to
an element in its range.
2
Note that:
a function maps any one element in its domain
to only one

Calculus and Vectors Problem Sheet for Week 1
Note: This problem sheet (for week 1) introduces some basic concepts in calculus and also serves to demonstrate
how problem sheets are to be used in the module. The problem sheet for any week is made available

3
MATH10121 A1
A1.1
Sets of Numbers
Number Sets. there are 5 basic number
sets1 :
A1.2
about innity ( )
innity is greater than any real number, i.e.
> x for any
N = cfw_1, 2, 3, the natural numbers
(positive whole numbers as used for counting)
Z = cfw_

MATH10121 Calculus and Vectors
Calculus is at the heart of much mathematics.
You must become very familiar with it. It will be used in most of the modules you study.
We expect you to know core A-level maths (indicated as Revision in notes).
Revise it all:

Sample Answers for Week 11
math 10121
Area Integrals
Easy Questions
ZZ
1
Z
1. (a)
2
Z
0
0
ZRZ
2
Z
0
1
Z
dy dx =
0
0
R
1
Z
(b)
0
x
Z
dy dx =
1
Z
0
0
0
R
1
Z
Z
R
Z
1
1
Z
dx dy =
Z
x
1
0
y
Z
dx dy =
0
ZZ
(d)
0
Z
0
Z
[y]0 dx =
x
[y]0 dx =
1
[x]y dy =
1
[y]x d

Sample Answer Sheet for Week 12
math 10121
Line Integrals
Easy Questions
1. (a) ~f = x3 + 4xy bi + 4xy y 3 bj is defined on the simply connected domain R2 .
~f = x 4xy y 3 y x3 + 4xy = 4y 4x 6 0, so ~f is not conservative.
(b) ~f = (ey , xey ) is defined

Problem Sheet for Weeks 67
math 10121
Integration and Curve Sketching
hand in starred questions for the supervision class in week 7
Easy Questions
1. Evaluate the following indenite integrals (you should be able to write down the answers immediately)
14

Problem Sheet for Week 8
math 10121
Vectors
Easy Questions
1. For each of the following pairs of vectors a and b and scalar c evaluate all of the quantities
|a|, |b|, a + b,
a b, c a, c b
and a b
and state whether or not a and b are orthogonal
(a) a = (1,

Sample Answer Sheet for Week 1
math 10121
Numbers and Functions
Easy Questions
1.
2.
(a)
x (4, 2)
(b)
x [ 2 , 2]
3
(c)
2x + y = 1
straight line with
y = 1 at x = 0
x = 1 at y = 0
2
(a)
(c)
(d)
x2 + 4y 2 < 4
inside of ellipse with
x = 2 at y = 0
y = 1 at x

Sample Answers for Week 5
math 10121
Integration
Easy Questions
(a)
sin d = cos + Const
(b)
cos x dx = sin x + Const
(c)
1.
sec2 r dr = tan r + Const
(d)
s13 ds =
(f)
1
du = ln |u| + Const
u
7
(e)
t1/7 dt = 7 t8/7 + Const
8
t dt =
s14
+ Const
14
(g)
ln v

Sample Answer Sheet for Week 3
math 10121
Limits and Derivatives
Easy Questions
(a)
6x(1 x2 )2
(b)
(e)
1.
3u2 + 1
2 u3 + u
(f)
2 sin t cos t
et cosh t sinh t
(a) y = 1 (x + 1)
4
3. (a)
d
(1 + x4 )
dx
(b)
t
(c)
(d)
(h)
3x2 exp(2x)
2 tan + 2 sec2
2z 3 ln

Problem Sheet for Week 12
math 10121
Line Integrals
Easy Questions
1. Determine whether or not f is a conservative vector eld. If it is, nd a function f such that f =
3
3
y
f
y
(a)
f = x + 4xy i + 4xy y j
(b)
f = (e , xe )
(c)
f = (y, x + 2y)
(c)
f = 1 ye

Problem Sheet for Week 11
math 10121
Area Integrals
Easy Questions
1. The double integral
ZZ
dx dy, which can also be written as
R
ZZ
1 dx dy, gives the area of the region R.
R
Use double integrals to nd the area of each of the following regions
(a) the r

Problem Sheet for Week 10
math 10121
Partial Derivatives
Easy Questions
1. If f is a scalar-valued function and F is a 3 dimensional vector-valued function, which of the following
expressions are meaningful in 3 dimensions. If so, state whether the result

Problem Sheet for Week 5
math 10121
Integration
Easy Questions
1. Provide formulae for the following indenite integrals
(c)
sec2 r dr
(f)
1
du
u
sinh w dw
(i)
cosh x dx
(k)
ez dz
(l)
(n)
(a)
sin d
(b)
(d)
s13 ds
(e)
(g)
ln v dv
(h)
(j)
sech2 y dy
1
ds
s2

Sample Answer Sheet for Week 10
math 10121
Partial Derivatives
Easy Questions
1. The operations curl and div are defined for a vector-valued function (say F); curl F = F is a
vector-valued function in 3 dimensions and div F = F is a scalar-valued function

math 10121
Problem Sheet for Week 10
Partial Derivatives
Easy Questions
1. If f is a scalar-valued function and F is a 3 dimensional vector-valued function, which of the following
expressions are meaningful in 3 dimensions. If so, state whether the result

MATH10121 A4
A4.1
A4.2
derivatives of polynomials
power series
Exercise 1.
A power series is a polynomial of infinite order
if p(x) = a0 + a1 x + a2 x2 + a3 x3
what are the values of p(0) and the
first three derivatives of p(x) at x = 0 ?
p(x) = a0 + a1 x

MATH10121 B1
points and vectors
B1.1
Revision
points in 3 dimensions
Real numbers are equivalent to points on a real line
a point x R is a distance x to the right of 0.
Cartesian representation in 3 dimensions:
A point (x, y, z) R3 , is
Revision
R3 is th

MATH10121 A2
A2.1
Functions
sketching some simple functions
We can say that
A2.2
Revision
1 t2 2t
= 2 (t + 1)2
x2
a function maps any
element in its domain to
an element in its range.
2
Note that:
a function maps any one element in its domain
to only one

math 10121
Problem Sheet for Week 9
Vectors & Partial Derivatives
Easy Questions
1. Give all first and second-order partial derivatives of the following functions:
(a)
f (p, q) = sin(p
(c)
h(t, u) = et
2
(e)
v(x, y) = xy
u
q)
(b)
g(r, s) = r2 + 2rs
(d)
u(

Calculus and Vectors Problem Sheet for Week 1
Note: This problem sheet (for week 1) introduces some basic concepts in calculus and also serves to demonstrate
how problem sheets are to be used in the module. The problem sheet for any week is made available

math 10121
Problem Sheet for Week 5
Integration
Easy Questions
1. Provide formulae for the following indefinite integrals
Z
Z
(a)
sin d
(b)
cos x dx
Z
Z p
7
(d)
s13 ds
(e)
t dt
Z
Z
(g)
ln v dv
(h)
sinh w dw
Z
Z
(j)
sech2 y dy
(k)
ez dz
Z
Z
1
1
p
p
(m)
ds