(Advance note: I'm not looking for the answers to this question. I want to understand what "the magnitude of the rate of change of $\theta$ increases without bound" means.)

Here's the problem I'm given: A boat is towed toward a dock by a cable attached to a winch that stands 14 ft above the water level. Let $\theta$ be the angle of the elevation of the winch and let L be the length of the cable as the boat is towed toward the dock.

Find $\lim_{L\to 14+} \frac{d\theta}{dL}$

Since I'm not including a drawing, let me make it clear that sin$\theta$ = $\frac{14}{L}$

I've worked through this problem and can get the correct answers. But, what in the world does it all mean?

Specifically, at the end, when I determine the limit is -$\infty$, this apparently means that, as the last foot of cable (L) is reeled in, the magnitude of the rate of change of $\theta$ increases without bound. What does this mean exactly in real world terms? Any insight is appreciated.


It means that as you reel in the last bit of cable, you stop towing and start lifting vertically. Near that point, for a very small amount of extra cable wound, there is a large change in the magnitude of the angle of elevation (by magnitude, read absolute value).

Draw some diagrams with $L=50,51$ versus $L=15, 15.1$ and let $\Delta L=1$.


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