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SIT194 Introduction to Mathematical Modelling 2018 – Paper 1
Trimester 2, 2018
Special Instructions
This examination is CLOSED BOOK.
Calculators are NOT ALLOWED.
Writing time is 2 HOURS.
This examination constitutes 60% of your assessment in this unit and is out of 60 marks in total.
You can use any results in the exam formula sheet but otherwise all steps (workings) to arrive at the
answer must be clearly shown.
The formula sheet (2 pages) is included at the back of the paper.
This examination question booklet must be handed in with any used answer booklets.
SIT194 Introduction to Mathematical Modelling Paper 1 2018 Tri 2 Page 2 of 6
Q.1) (a) Determine whether the function
f(x) =
x3 − 4x
x4 + 2x2
is even, odd, or neither.
(b) For the function y = f(x) = |x− 2|, sketch the graph (showing all required working) and find a
restriction of the domain such that the function is one-to-one.
(c) Evaluate the following limits.
(i)
lim
x→−3
x2 − 3x− 18
x2 + 8x+ 15
(ii)
lim
x→∞
6x3 − 4x+ 7
5− 2x2 − 3x3
(iii)
lim
x→0
e2x − 1− 2x
x2
[2+2+(2+2+2) = 10 marks]
Q.2) (a) Find dydx for each of the following:
(i) y = (xex + sinx)7
(ii) y = (x3 + x) sin−1 x
(b) Use logarithmic differentiation to find dydx when
y =
(x4 − 3x3 + 5)1/3
(x3 − 7x2)2/5
(c) Use implicit differentiation determine dydx if
(xy + 1)2 =
(y2 + 3)1/2
(y3 − 1)1/3
[(2+2)+3+3 = 10 marks]
SIT194 Introduction to Mathematical Modelling Paper 1 2018 Tri 2 Page 3 of 6
Q.3) (a) Use substitution to find
I =
∫
x2(x3 − 6)1/2 dx
(b) Use integration by parts to find
I =
∫
(x+ 2)e−x dx
(c) Use standard integrals to evaluate of the following integral
I =
∫
1
(x2 + 9)
dx
(d) Use partial fractions to find
I =
∫ −3x+ 23
x2 + x− 12dx
[2+3+2+3=10 marks]
Q.4) (a) Find the volume formed when the area, on the positive x-axis that is below the curve
y = (1− 2x) and above the x-axis, is rotated about the x axis.
(b) Solve the following differential equation
dy
dx
= xy; y(0) = 3
(c) Solve the following differential equation
dy
dx
+
2y
x
= x; y(1) = 2
(d) Use the ratio test to determine whether the following series converges or diverges
∞∑
n=1
n3
e2n
.
[3+2+3+2= 10 marks]
Q.5) (a) Use established MacLaurin series to find the first three non-zero terms of
f(x) = e−x
2/2
(b) Use your expression in (a) to find an approximation of
a∫
0
x2e−x
2/2 dx,
provided a is within the interval of convergence. Express your approximation as a function of a.
[3+3=6 marks]
SIT194 Introduction to Mathematical Modelling Paper 1 2018 Tri 2 Page 4 of 6
Q.6) (a) Find the cosine of the angle between the two vectors
˜
A = 2
˜
i−
˜
j + 3
˜
k,
˜
B =
˜
i+ 2
˜
j + 3
˜
k
(b) Find a unit vector that is perpendicular to the two vectors
˜
A = 3
˜
i−
˜
j + 2
˜
k,
˜
B = 2
˜
i+
˜
j − 2
˜
k
(c) Find parametric equations for the line through the point P (4,−6,−3) and parallel to the vector
˜
v =
˜
i− 3
˜
j − 5
˜
k
[2+3+3=8 marks]
Q.7) Harder questions
(a) Is there a number b such that
lim
x→1
x2 + 2bx− b− 2
x2 − 4x+ 3
exists? If so, find the value of b and the value of limit.
(b) Evaluate the following integral
I =
∫
e−3x cos(2x) dx
[3+3=6 marks]
SIT194 Introduction to Mathematical Modelling Paper 1 2018 Tri 2 Page 5 of 6
SIT194 EXAM Formula Sheet - Integrals and formulae [2 pages]
Trigonometric Identities
1. sin(−θ) = − sin(θ) 5. cos(2θ) = 2 cos2(θ)− 1
2. cos(−θ) = cos(θ) 6. cos(2θ) = 1− 2 sin2(θ)
3. cos2(θ) + sin2(θ) = 1 7. cos2(θ) = 12(1 + cos(2θ))
4. sin(2θ) = 2 sin(θ) cos(θ) 8. sin2(θ) = 12(1− cos(2θ))
Integrals of Elementary Functions
1.
∫
xn dx = x
n+1
n+1 + C, n 6= −1 5.
∫
cosx dx = sinx+ C
2.
∫
1
x dx = ln |x|+ C 6.
∫
sec2 x dx = tanx+ C
3.
∫
ex dx = ex + C 7.
∫
sinhx dx = coshx+ C
4.
∫
sinx dx = − cosx+ C 8. ∫ coshx dx = sinhx+ C
Standard Integrals
1.
∫
1
a2−x2 dx =
1
a tanh
−1 (x
a
)
+ C or 12a ln
a+x
a−x + C if |x| < a
2.
∫
1
x2−a2 dx = − 1a coth−1
(
x
a
)
+ C or 12a ln
x−a
x+a + C if |x| > a
3.
∫
1
x2+a2
dx = 1a tan
−1 (x
a
)
+ C
4.
∫
1√
a2−x2 dx = sin
−1 (x
a
)
+ C or − cos−1 (xa)+ C
5.
∫
1√
x2−a2 dx =
 cosh
−1 (x
a
)
+ C or ln
[
x+
√
x2 − a2
]
+ C, x > a
− cosh−1 (−xa )+ C or ln [−x+√x2 − a2]+ C, x < −a
6.
∫
1√
x2+a2
dx = sinh−1
(
x
a
)
+ C or ln
[
x+
√
x2 + a2
]
+ C
Integration by parts ∫
u dv = uv −
∫
v du
Hyperbolic functions
sinhx =
ex − e−x
2
coshx =
ex + e−x
2
tanh =
sinhx
coshx
Fundamental Theorem of Calculus
d
dx
 x∫
a
f(t) dt
 = f(x), d
dx
 u(x)∫
a
f(t) dt
 = f(u)du
dx
.
SIT194 Introduction to Mathematical Modelling Paper 1 2018 Tri 2 Page 6 of 6
L’Hoˆpital’s Rule
If lim
x→a
f(x)
g(x)
=
0
0
or
±∞
±∞ , then limx→a
f(x)
g(x)
= lim
x→a
f ′(x)
g′(x)
.
Differential Equations
Separable
dy
dx
= g(x)f(y) Linear
dy
dx
+ p(x)y = q(x), integrating factor µ(x) = e
∫
p(x) dx
Solids of Revolution
V = pi
b∫
a
h2 dx, where h(x) is a typical circular radius.
Convergence of Sequences and Series
Sequences {an} is convergent if lim
n→∞ an exists (i.e. unique and finite)
Geometric Series
∞∑
n=1
arn−1 is convergent with sum S = a1−r for |r| < 1, divergent for |r| ≥ 1
p-Series
∞∑
n=1
1
np is convergent for p > 1, divergent for p ≤ 1
nth term test If lim
n→∞ an 6= 0, then
∞∑
n=1
an is divergent
Ratio test If lim
n→∞
an+1
an
< 1, then the series is convergent, divergent if lim
n→∞
an+1
an
> 1
Power series
∞∑
n=0
bn(x− c)n =
∞∑
n=0
an is convergent when lim
n→∞
∣∣∣an+1an ∣∣∣ < 1
MacLaurin Series
f(x) =
∞∑
n=0
1
n!
f (n)(0)xn
Established MacLaurin Series
ex =
∞∑
n=0
xn
n! ; −∞ < x <∞ 11−x =
∞∑
n=0
xn; −1 < x < 1
sinx =
∞∑
n=0
(−1)nx2n+1
(2n+1)! ; −∞ < x <∞ cosx =
∞∑
n=0
(−1)nx2n
(2n)! ; −∞ < x <∞
Vectors
For
˜
A = a1
˜
i+ a2
˜
j + a3
˜
k and
˜
B = b1
˜
i+ b2
˜
j + b3
˜
k
˜
A ·
˜
B = |
˜
A||
˜
B| cos θ = a1b1 + a2b2 + a3b3
˜
A×
˜
B =
∣∣∣∣∣∣ ˜
i
˜
j
˜
k
a1 a2 a3
b1 b2 b3
∣∣∣∣∣∣