SIT194: Introduction To Mathematical Modelling Assignment 1
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SIT194: Introduction To Mathematical Modelling
Assignment 1 (17% of unit)
Due date: 8:00pm AEST Thursday, 3 August 2023
Questions
1. Determine if the following functions are even, odd or neither.
(i) f(x) = 
(ii) f(x) = 
(iii) f(x) = cosh（ee2x(x/2)
e(e)
2x(x/)2 ） (6 marks)
2. For the function y = f(x) = | − 2x + 4| ,
(i) Clearly sketch the function showing important points, i.e. intercepts.
(ii) Determine the domain and range of the function.
(iii) Find a restriction of the domain such that the function is one-to-one. (3 marks)
3. Evaluate the following limits:
(i)
x
1 x2(x2) 
7x(2x) 
6(3)
(ii)
x(l)
6(2)
7(+)x
5+−21(6)x(1)
(iii)
x(l)
e(s)x(2)
x5 (6 marks)
4. Find the derivative of the following functions:
(i) y = (xcoshx + cosx)4
(ii) y = (e2x − 6 √x)sin−1 x (4 marks)
5. Using implicit differentiation, determine dx(dy) if
sin(ye−x ) = y3 + cosx (3 marks)
6. Using logarithmic differentiation, determine dx(dy) , in terms of x, if
(i) y = 
(ii) y = 
(iii) y = (x5 cos2(−6x)2
4 (6 marks)
7. *Consider a circle in the x− y Cartesian plane. Determine the radius a, and centre coordinates (h, k), of the circle that satisfies the following properties:
❼ the(Th)epa(ci)r(r)a(c)b(le)ola(is)tat(an)
2(e)
1(t))to the parabola y2 = x − 1 at the point (2, 1),i.e. the circle touches
❼ The second derivatives, i.e. dx(d2)
, of both the circle and parabola have the same value at
the point (2, 1).
Note that d(d)x(2)
= dx(d)（ dx(dy)）. (6 marks)
2023-08-12