Activity 2.3.3.
Use Figure 2.3.12 in the text (which plots the sine and cosine functions ont the same axes) to assist in answering the following questions.
(a)
Give an example of the largest interval you can find on which \(f(t) = \sin(t)\) is decreasing.
(b)
Give an example of the largest interval you can find on which \(f(t) = \sin(t)\) is decreasing and concave down.
(c)
Give an example of the largest interval you can find on which \(g(t) = \cos(t)\) is increasing.
(d)
Give an example of the largest interval you can find on which \(g(t) = \cos(t)\) is increasing and concave up.
(e)
Without doing any computation, on which interval is the average rate of change of \(g(t) = \cos(t)\) greater: \([\pi, \pi+0.1]\) or \([\frac{3\pi}{2}, \frac{3\pi}{2} + 0.1]\text{?}\) Why?
(f)
In general, how would you characterize the locations on the sine and cosine graphs where the functions are increasing or decreasingly most rapidly?
(g)
Thinking from the perspective of the unit circle, for which quadrants of the \(x\)-\(y\) plane is \(\cos(t)\) negative for an angle \(t\) that lies in that quadrant?
