Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 497

cation that the height y is an area where climbers are likely to drop 17 out.
Perhaps the wall at that height is a challenging area. S ′ (y) closer to zero is
an indication that fewer climbers drop out at height y. Perhaps there is a
plateau that climbers can reach, and there they rest. We might expect S ′ (y)
to decrease just after this plateau, since the climbers are more rested. In
fact, S ′ (y) might also become closer to zero just before the plateau, because
as climbers begin to get close to this milestone, they push a bit harder and
are more reluctant to give up. 18
Figure 11.6
Happiness
versus salary
One last example. Figure 11.6 shows happiness as a function of salary.
In this case, the derivative is essentially the same thing as what economists
would call “marginal utility.” It's the ratio of additional units of happiness
per additional unit of income. According this figure, the marginal utility
of income decreases, which of course is the famous law of diminishing re-
turns. According to our research, 19 it even becomes negative after a certain
point, where the troubles associated with high income begin to outweigh
the psychological benefits. The economist-speak phrase “negative marginal
utility” is translated into everyday language as “stop doing that.”
17 So to speak.
18 The alert reader may have noticed that, because there are only a finite number
of climbers to sample, S(y) comes from samples data, and therefore has discontinuous
“steps” in it, rather than being a smooth line. At those steps, the derivative isn't really
defined. For purposes of this illustration, let's assume that we have fit a smooth curve
through the experimental data in order to yield a continuous function.
19 We just made quotes in the air with our fingers as we wrote that word.