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MATHS 102 - Functioning in Mathematics

Department of Mathematics

Assignment 2


1. (a) (i) Briefly explain the error involved in simplifying to:

[2 marks]

(ii) Hence find all solutions to ,  [4 marks]

(b) Consider the equation, ,

(i) Find the principal angles and the period required to solve this equation (you need to show working to get full marks). [3 marks]

(ii) Hence solve the equation;  [3 marks]

(c) Consider the equation, ,

(i) Find the principal angles and the period required to solve this equation (you need to show working to get full marks). [3 marks]

(ii) Hence solve the equation; ,  [3 marks]

(d) A 400 meter(m) athletics track has two semi-circular ‘bends’. In all track events, the athletes run in a counter clockwise direction. When measured along the inside (left) margin of lane 1, each semi-circle has a length of 115.611 metres (3dp). The straight sections of the track each measure 84.39 metres.

(i) Give the length of the radius to the inside margin of lane 1 for each semi-circular section (accurate to 2 decimal places). [2 marks]

(ii) In this question we will consider the 200 m sprint event. There are 8 lanes, and each lane has a width of 1.22 metres.

Using the idea of arch-length, show how you would compute the distance ahead of lane 1 that the athlete in Lane 8 (accurate to 2dp) must start so they both run the same distance (assume each sprinter runs on the inside margin of their lane). This is called the “200 metre stagger for lane 8”. [3 marks]


2. (a) A causeway joining an island to the mainland is flooded twice a day due to tidal flow. On a particular day, the tide is on its way in and is 4.16 metres above sea level at 4:30am. The causeway is opened for traffic 3hrs after high-tide, and all traffic must be off the causeway 3hrs before the next high-tide. In this problem we will assume that for the week during which the day we are looking at occurs, the tidal flow is modelled by the formula:

where h(t) is the height of the tide recorded in metres above sea level at time t measured in hours.

NOTE: This question is written in the context of a fine summer day in a region where there is an average of 17.5 hours daily sunlight in the summer period. Sea level refers to mean sea level, that is the sea level that would be recorded in the absence of tides.

(i) What is the period - and what is its practical significance? [2 marks]

(ii) Give the values for high-tide and low-tide [2 marks]

(iii) At what time does the high-tide and the following low-tide occur? [4 marks]

(iv) What is the approximate height of the causeway above sea level? [2 marks]


3. (a) Suppose we have a composite function h(x) = f(g(x)). Give a brief explanation why the range of g(x) must be checked when finding the domain of h(x). [2 marks]

(b) For each of the following find their domains.

(i) (f ○ g)(x) where  and  [3 marks]

(ii) (h ○ k)(x) where  and  [3 marks]

(c) Given , and , show that (f  g)(x) = (g f)(x) and briefly explain what this means. [5 marks]


4. Find the derivative using First Principles, that is, the definition:

 (ref: Pages 137-140 Coursebook) [4 marks]

Total Marks: 50