MAT2691/101/0/2025

Tutorial letter 101/0/2025

MATHEMATICS II (ENGINEERING)
MAT2691 Assignment 01

Year module

Department of Mathematical Sciences

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university
Define tomorrow. of south africa
 MAT2691/101/0/2025

Question 1: 37 Marks

dy
Find for the following functions in their simplest form. Use trigonometric identities to simplify where
dx
possible.

1
(1.1) y=√
3
(3)
1 + x + sin x
√
(1.2) y= x2 + 1 (2)

y
(1.3) tan(x − y) = (4)
1 + x2

(1.4) x 3 y = (1 + x 2 )x (5)
r
3x 4 sin x

x
(1.5) y = ln (5)
e2x sec x

(1.6) sin(x + y) + cos(x + y) = 12 + y + x 4 (5)

3 √
x4 cos x + sin x
(1.7) y= (5)
(4x + 2)5

(1.8) y = tanh−1 (sin x) (3)

r 
1−x
(1.9) y = arctan (5)
1+x

Question 2: 14 Marks

d 2y
Find in its simplest form
dx 2

(2.1) If x = b sin 3t and y = b cos 3t. (4)

(2.2) If x 4 + y 4 = 16 (5)

1
(2.3) y= (5)
(1 + tan x)2

Question 3: 17 Marks
 
x −y
If z = sin , determine the following
x +y

∂z
(3.1) (3)
∂x
 ∂z
(3.2) (3)
∂y

∂ 2z
(3.3) (5)
∂x∂y

∂ 2z
(3.4) (5)
∂y∂x

∂ 2z ∂ 2z
(3.5) What conclusion can be drawn about and ? (1)
∂y∂x ∂x∂y

Question 4: 4 Marks

In structural engineering, Euler’s formula is used to determine the critical axial load that a long, slender
column can carry without buckling. The formula is given by:

π 2 EI
L=
h2
where E is the modulus of elasticity, I is the second moment of area (moment of inertia), and h is the column
length.

(4.1) If the length h of the column is increased by 3% and the moment of inertia I is increased by (4)
5%, determine the percentage change in the axial load L. (4)

(4.2) Will the column become more stable or less stable? Justify your answer.

Question 5: 8 Marks

(5.1) Use the Maclaurin series to expand ex to four non-zero terms. (3)

(5.2) Use the Maclaurin series to expand e−x to four non-zero terms. (3)

(5.3) Use your answers in question 5.1 and 5.2 to write down a series expansion for cosh x to four (2)
non-zero terms.

TOTAL 80