Answer to Question #97346 in Calculus for Matt Maffey

Question

The displacement of a mass is given by the function y=sin3t

The displacement of a mass is given by the function y=sin3t

a) draw a graph of the displacement y(m) against time t (s) for the time t= 0s to t=2.s?

b) identify for 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?

d) compare the results from part b and c?

Answer 1

Solution:

We are going to draw the graph of the displacement and also maxima and minima

a) The graph of the displacement y(m) against time t (s) for the time t= 0s to t=2.s

b) We can get the maximum and minimum from the graph

Maximum at x = 0.52 or ($\frac {\pi} {6} )$

Minimum at x = 1.57 or ($\frac {\pi} {2})$

(c) To find the maximum and minima, we need to find the $y\prime$ and set this to zero.

$$y = sin 3t$$

differentiate with respect to x

$$y\prime = 3 cos 3t = 0$$

$$3t = \frac {\pi}{2}, \frac {3\pi}{2}$$

$$t = \frac {\pi}{6}; \frac {\pi}{2} = 0.52; 1.57$$

Now,

$$y\prime\prime = - 9 sin 3t$$

$y\prime\prime (0.52) < 0\\ So, y \space has \space maximum \space at \space t = 0.523$

To find the Inflection point set $y\prime\prime = 0$

$- 9 sin 3t = 0\\ sin 3t = 0 \space at \space 3t = \pi$ or 3.14

inflection point is at $t = 1.04$

(d) The points of maximum and minimum in two methods are same.