Find the angle at the top corner of the teardrop curve defined by
the equation √(x2+y2)=ey-1
Graphic look.
https://www.desmos.com/calculator/fq7krtyxfa
or zoomed in on (0,1)
https://www.desmos.com/calculator/97763u0t3c
Visually, sure looks like 90 degrees
It is at (0,1) that we want to know the slope.
Taking the derivative is probably what is needed, but I'll try something else.
I think the angle is 90 degrees and therefore the slope is ±1.
If the slope is 1, and (0,1) is on the curve then (ε, 1+ε) is also on it.
Plugging (0,1) to the equation gives 1=1. Check.
Try plugging (ε, 1+ε):
√(1 + 2ε + 2ε^2)=e^(ε)
1 + 2ε + 2ε^2 = e^(2ε)
Taylor expansion first 3 terms for e^(2ε) is:
1 + 2ε + (2ε)^2/2!
1 + 2ε + 2ε^2 so this checks.
So the slope is 1 and by symmetry -1 for the other limb; therefore the angle is 90 degrees.
Not a formal proof but a handwaving argument.
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Posted by Larry
on 2025-05-31 21:04:21 |
Probable answer; some hand waving