Mathematics

Applied Mathematics
Study Period 2, 2021
Assignment 2
Due by: 4:00pm Tuesday 13 July
1. [22 marks] Using the method outlined in Section 5.4 of the textbook (also described in
lectures), and showing the working, sketch the graph of each of the following functions.
Note: you do not need to consider f
00(x), concavity and points of inflection (step 6).
(a) f(x) = 2x
2
(x − 1)(x + 3)
(b) f(x) = 4x
2−9
x2−1
2. [8 marks] Find the absolute maximum and minimum of the function
f(x) = (2x
2 + 1) x
4/3
for x ∈ [−1, 8]
Express your answers in simple exact form.
3. [8 marks] A company wants to manufacture an open cylindrical bucket of volume 10 litres
(10000 cm3
). The plastic used for the base of the bucket costs 0.08 cents per cm2 while
the plastic used for the curved side of the bucket costs 0.03 cents per cm2
. Find the
radius and height of the bucket for which the bucket has minimum cost. What is the
minimum cost? Show all the reasoning and evaluate your answers to 2 decimal places.
4. [7 marks]
(a) Find dy/dx for 2x
2y
2 +
√
3x + 7y = 22
(b) Show that the point (x, y) = (3, 1) lies on the curve defined by the equation in part
(a), and find the slope of the tangent line at this point.
5. [8 marks] Granular sugar is leaking out of a container at a chocolate factory and forming
a conical pile on the floor. The conical pile always has a base width 1.5 times the height
of the pile. If the base radius of the pile is increasing at rate 3cm/s when the base radius
is 90cm, at what rate is the sugar leaking out of the container? (Volume of cone of height
h and base radius r is V =
1
3
πr2h.)
6. [7 marks] Let f(x) = −3 + (3x
2 − x + 1) ln(2√
x − 5)
(a) Find the derivative of f.
(b) Using the derivative and linear approximation, estimate f(9.1).
Notes:
• 5 marks are reserved for presentation in this assignment.
• There are penalties for late assignments. You must contact your tutor before the due
date if you have difficulties making the deadline.