During one of my FP1 past-papers, of which I have been trawling through the entire lot, it struck me that there was one part of the course that was very much a matter of "here's the result, learn it". Hyperbolas are only touched-upon in the AS further maths specification, and for me that makes the given theory more difficult to engage with. Hence, I found in one question that I could not remember the generalisation for the asymptotic gradient of a hyperbola. Well, with two options in mind, I thought asking for help somewhat non-proactive so I decided to see if I could work it out with a little bit of limit notation.
Having achieved this, feeling extremely pleased with my elevated understanding of the function despite having not blasted through the 3 or 4 AS level questions I would have otherwise spent the time with, I began to consider what would happen if the same general form of hyperbola was used with higher-order indices. My findings are on the A4 scans below.
The second page deals with more trivial cases, where x and y are raised to different powers - there produce fairly uninteresting polynomial-type curves, which could effortlessly be rearranged to express the first-quadrant branches explicitly. Nevertheless an implicit-differentiation method for proving the limits of the gradients is still fairly stimulating, and the result is pretty nice, if quite obvious.
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