MAS164: Fundamentals of Mathematics Semester 1, 2023 Assignment 4 - Due 5pm, Friday 12 May This assignment is worth 55 marks, with 5 marks reserved for presentation. Please note that marks will be deducted for solutions that are not properly presented, for working that is inadequate, or for untidy work. Always show full working. Where possible give the exact answer. 1. [14 marks] Consider the trigonometric function f(t) = -1 + 4sin(}a(t -1)). (a) What is the amplitude of f(t)? (b) What is the period of f(t)? (c) What are the maximum and minimum values attained by f (t)? (d) Sketch the graph of f(t) for te [-1, 3]. 2. [16 marks] A sound wave has frequency 50 Hz (i.e. it has wavelength (period) = 0.02 seconds). Within each beat, the highest sound intensity is 100 decibel (dB) and the lowest intensity is 60 dB. Supposing that t = 0 corresponds to the low intensity, write down a trigonometric function that describes this sound wave. Sketch the sound wave over two periods. (Hint: Since the period = 0.02 seconds which is very small, make the scale on the horizontal axis really big when you sketch the graph). 3. [8 marks] The volume of water in a tank at a particular time (measured in seconds) is given by V(t) = 5t(2 -t) m'. Find the rate of change of the volume of water in the tank from first principles (using the definition of the rate of change). 4. [12 marks] Consider the function f(x) = x- - 1. Find the equation of the tangent to the graph of f(x) at x = 3. [NOTE: when calculating f'(3), use first principles.] . Note: There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline