2.The displacement of a mass is given by the function y=sin3t The tasks are to: a) Draw a graph of the displacement y(m) against time t(s) for the time t=0s to t=2.s b) Identify the position of any turning points and whether they are maxima minima or points of inflexion. c) Calculate the turning points of the function using differential calculus and show which are maxima minima or points of inflexion by using the second derivative. Compare your results from parts b and c.

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2.The displacement of a mass is given by the function y=sin3t The tasks are to: a) Draw a graph of the displacement y(m) against time t(s) for the time t=0s to t=2.s b) Identify the position of any turning points and whether they are maxima minima or points of inflexion. c) Calculate the turning points of the function using differential calculus and show which are maxima minima or points of inflexion by using the second derivative. Compare your results from parts b and c.

Answer 1

This is a problem of mathematics that needs to be solved through calculation. To give you a rough idea, knowing the graph of a sinusoidal function, we can tell for part a) any sketches would generate a wave that oscillates with an amplitude of 1 and period of \(2\pi / 3\). To answer parts b) and c), turning points of the function would occur at times \(t = \pi / 6 + n\pi\)s, for \(n \) being any integer represents rounds. Confirming with calculus method, deriving funcition gives \(y'=3\cos{3t}\), equals to zero so \(t = \pi / 6 + n\pi\), agreeing with b).

Answer 2

The question is about trigonometry and calculus. It relates to a mathematical problem regarding the mapping of graph, identifying or calculating turning points and understanding of maxima, minima, and inflection points. a) To draw the plot, we use the information that \(y= \sin{3t}\). This is a sine function being plotted in the Cartesian plane. Graphing such a function would should help you visualize the behaviour of the change in \(y\), with time \(t\). b) Intuitively, from understanding the sine function, we can see that the turning points on the graph occur at the crest and trough of each wave - where sine hits its maximum and minimum values, which are 1 and -1 respectively. As an orthodox sine wave, we know it oscillates between these two extreme values. Both of these kinds of turning points can be seen in a single period of \(\sin x \), that is from 0 to \(2\pi\) before it starts to repeat. c) The turning points of a function can also be more rigorously calculated using calculus - particularly, it is when the derivative is 0. Taking the derivative of \(y\), we get \(y'=3\cos{3t}\). Setting \(y'\) equals to zero to find maximam and minimum, once we have derived a second derivative (the derivative of \(y'\)). The sign of the derivative provides the answer if the function whether reaches a maximum, minimum or inflection point. Let's do it step by step on a following calculation. The results are supposed to be compared in your side as explaning individual calculations for parts \(b \) and \( c \) as an output is not possible at the same momment without understanding the pattern and it's detail behaviour mathimatically.

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