MAS164:FundamentalsofMathematics

StudyPeriod2,2024

ASSIGNMENT3

1. [12marks]Considerthetrigonometricfunction f(t)=2+3sin(4?(t?1)).
   1. Whatistheamplitudeoff(t)? [2marks]
   2. Whatistheperiodoff(t)? [2marks]
   3. Whatarethemaximumandminimumvaluesattainedbyf(t)? [4marks]
   4. Sketchthegraphoff(t)fort?[?1,1]. [4marks] [12marks]A water wave in a lake has a frequency of 20 Hz (i.e.,a wavelength or period of0.05 seconds).Within each wave, the water level reaches a maximum height of 2.5 meters anda minimum height of 0.5 meters.Supposing that t=0 corresponds to the water level at its minimum height, write down a trigonometric function that describes this water wave.Sketch the water wave over two periods.(Hint: Since the period = 0.05 seconds, which is very small, make the scale on the horizontal axis really big when you sketch the graph).

2.[12 marks] A water wave in a lake has a frequency of 20 Hz (i.e., a wavelength or period of 0.05 seconds). Within each wave, the water level reaches a maximum height of 2.5 meters and a minimum height of 0.5 meters. Supposing that t = 0 corresponds to the water level at its minimum height, write down a trigonometric function that describes this water wave. Sketch the water wave over two periods. (Hint: Since the period = 0.05 seconds, which is very small, 3. [22 marks] Evaluate the following limits (using calculus and showing all your working) @ lim, ,,, o= [3 marks] (b) limy-o 322 [3 marks] (c) limy o 25661 [4 marks] (d) Tim,-> 22 [4 marks] (e) limys f;'i'gg [4 marks] (0) limsoo 22 (4 marks] 4. [10 marks] The position of a car moving along a straight road at time t (in seconds) is given by the function S(#) = (52 + 10f) meters. Find the velocity of the car at time t using first principles (using the definition of velocity as the rate of change of position with respect to time). 5. [14 marks] Consider the function f(x) = x* + 3. Find the equation of the tangent to the graph of f(x) at x = 3. [NOTE: when calculating f.(3), use first principles.] * There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline