- Show that the equation $\sin x - \frac{1}{ 2}x = 0$ has a root in
$ [1,2] $.
- Is the root in $ [1,1.5] $ or in $ [1.5,2.0] $?
- Continue in this manner until you have located the root in an interval
of length $1/8$. {(Hint: Recall the Intermediate Value Theorem.)}
This problem isn’t that hard as stated, but can be taken up a notch in difficulty by asking in (a) that sin x = x/2 somewhere in [1,2]. I would also rephrase it (b) is there a root in [1,1.5] or [1.5, 2.0], since there in theory could be multiple roots on that interval.
The subtle difficulty in this is for students to estimate sin 1 > 1 (not necessary to estimate sin 2 since it’s clearly less than 1). This is obvious if you go back to the unit circle, but may be harder if you rely on a calculator and are now deprived of it, or may be longer if you try to reconstruct sine from a graph.
– Eric Hsu 1998-08-09