Activity 2.1.4.
Consider the same setting as Activity 2.1.3: a weight oscillates back and forth on a frictionless table with distance from the wall given by, \(h = f(t)\) (in inches) at any given time, \(t\) (in seconds). A graph of \(f\) and a table of select values are given below.
| \(t\) | \(f(t)\) |
| \(0.25\) | \(6.087\) |
| \(0.5\) | \(4.464\) |
| \(0.75\) | \(3.381\) |
| \(1\) | \(3.000\) |
| \(1.25\) | \(3.381\) |
| \(1.5\) | \(4.464\) |
| \(1.75\) | \(6.087\) |
| \(2\) | \(8.000\) |
| \(t\) | \(f(t)\) |
| \(2.25\) | \(9.913\) |
| \(2.5\) | \(11.536\) |
| \(2.75\) | \(12.619\) |
| \(3\) | \(13.000\) |
| \(3.25\) | \(12.619\) |
| \(3.5\) | \(11.536\) |
| \(3.75\) | \(9.913\) |
| \(4\) | \(8.000\) |
(a)
Determine \(AV_{[2,2.25]}\text{,}\) \(AV_{[2.25,2.5]}\text{,}\) \(AV_{[2.5,2.75]}\text{,}\) and \(AV_{[2.75,3]}\text{.}\) What do these four values tell us about how the weight is moving on the interval \([2,3]\text{?}\)
(b)
Give an example of an interval of length \(0.25\) units on which \(f\) has its most negative average rate of change. Justify your choice.
(c)
Give an example of the longest interval you can find on which \(f\) is decreasing.
(d)
Give an example of an interval on which \(f\) is concave up. (Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up.)
(e)
On an interval where \(f\) is both decreasing and concave down, what does this tell us about how the weight is moving on that interval? For instance, is the weight moving toward or away from the wall? is it speeding up or slowing down?
(f)
What general conclusions can you make about the average rate of change of a circular function on intervals near its highest or lowest points? about its average rate of change on intervals near the function’s midline?
