1. The absolute value occurs from the symmetry of the function across the -axis:
2. Most of these problems appear to relocate the denominator of the numerator into the denominator, as in
For example, in (a),
3. In the second step, we use the rule that
together with the fact that
4. A horizontal asymptote occurs when attains a finite value.
5. The book may have simplified to a fraction in order to determine more easily when the value of the derivative was positive or negative. In particular,
is positive when both the numerator and denominator have the same sign, and negative when they have differing sign.
6. Because the denominator of is never , we can multiply it out in the equation
to obtain an equivalent statement
The solutions here are just those values of , in radians, for which the corresponding point on the unit circle lies on the -axis: