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PHAS0002
Mathematical Methods I
Attempt all questions from both sections.
The numbers in square brackets show
the provisional allocation of maximum marks
per question or part of question.
Section A
[Part marks]
1. (a) State the formal definition of the derivative of a function f (x). [3]
(b) Using the formal definition of a derivative, calculate from first [3]
principles the derivative of f (x) = 3x2 − 7.
2. (a) The sum of an arithmetic series can be written as [3]
Sn =
n−1∑
k=0
(a + kd).
Show that this adds to
Sn =
n
2
(2a + (n − 1)d).
(b) A frog is at the bottom of a well. He finds that he can jump up
the side of the well, hanging on briefly between jumps. The
procedure is exhausting so he jumps a shorter distance each
time, starting with 1 m, then 12 m,
1
3 m, and so on. How deep [4]
must the well be for him never to escape, or will he always
gain his freedom? Briefly explain your answer.
PHAS0002/2019 TURN OVER
1
[Part marks]
3. (a) Given a general differential, df , of the form: [2]
df = A(x , y )dx + B(x , y )dy
state the condition that means that df is exact.
(b) Hence determine whether the following are exact differentials:
i. df = y tan x dx + x tan y dy ; [2]
ii. df = y (1 + x − x2)dx + x(x + 1) dy ; [2]
iii. df = 2xy ln x dy + y2(ln x + 1) dx . [2]
4. (a) Write down the general form of the Taylor series for a function
f (x). [2]
(b) Determine the Taylor series up to the cubic power of the
following:
i. ln x , about x = 1; [3]
ii. tan x , about x = pi. [3]
PHAS0002/2019 CONTINUED
2
[Part marks]
5. Calculate the following limits:
(a)
lim
x→+∞
[
2x2 + 1
4x2 + 3x + 1
]
,
[3]
(b)
lim
θ→pi/2
[
cos θ
θ2 − pi2/4
]
.
[3]
6. (a) Find the real and imaginary parts of [2]
−4− 3i
3 + 4i
.
(b) Use an Argand diagram to represent the complex number
z = −4− 3i , and write it in polar form. [3]
PHAS0002/2019 TURN OVER
3
Section B
[Part marks]
7. (a) Calculate the first derivative of the following functions:
i.
f (x) =
√
x√
x − 1
[2]
ii.
g(x) = exp
[
ln
(
x2
)]− 3x−7
[2]
(b) Evaluate the following integrals:
i. ∫
x3 ln xdx
[3]
ii. ∫
x
x2 − 3x − 4dx
[3]
iii. ∫ +1
−1
x exp(−|x |)dx
[3]
(c) A vector field F is given by
F = xy2 iˆ + 2 jˆ + x kˆ,
and C is a path parameterized by x = 4t , y = 4/t and z = 5, [7]
for the range 1 ≤ t ≤ 2. Evaluate the line integral∫
C
F · dr.
PHAS0002/2019 CONTINUED
4
[Part marks]
8. (a) For two vectors, a and b, define the scalar and vector [3]
products in terms of the magnitudes of the vectors and angle
between the vectors.
(b) For two vectors, a = 3iˆ + 2jˆ− 9kˆ and b = iˆ− 2jˆ− 4kˆ, [3]
determine the angle between them using both scalar and
vector products.
(c) Find the perpendicular distance, d , from the point P = (1, 1, 1) [6]
to the line, L, which passes through the points,
P1 = (−3, 1− 4) and P2 = (4, 4,−6) .
(d) The magnetic induction, B , is defined by the Lorentz force [8]
equation
F = q(v× B) .
Carrying out two experiments, we find
v = iˆ,
F
q
= 2kˆ− 4jˆ ,
v = jˆ,
F
q
= 4iˆ− kˆ .
From the results of these two experiments, calculate the
magnetic field induction, B , and verify that it agrees with a
third experiment where
v = kˆ,
F
q
= jˆ− 2iˆ .
PHAS0002/2019 TURN OVER
5
[Part marks]
9. (a) Given a function z = f (x , y ), state the condition for a point [3]
(x0, y0) to be stationary.
(b) Given a function z = f (x , y ), state the criteria to determine the [5]
nature of a stationary point.
(c) Find the stationary point(s) of f (x , y ) = x3 + 3y − y3 − 3x and
discuss its/their nature. [6]
(d) Use the method of Lagrange Multipliers to find the stationary
points of f (x , y , z) = x + 2y − 2z subject to the constraint
x2 + y2 + z2 = 1. [6]
