The graphs of the triad #y = cosh x, y = sinh x and y = e^x# meet x = 1 at (1, cosh 1), (1, sinh 1) and (1, e). How do you find these points for the FCF-triad #y=cosh(x(1+1/y)), y=sinh(x(1+1/y)) and y=e^(x(1+1/y))#?

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Answer 1 · 2016-09-06T07:53:45

See the explanation.

My long precision approximations to the y-coordinates of these

points are

2.241188143132744

2.082890025968637

3,591121476668622.

I hope that you get the same.

Of, course, you can verify this by substitution.

These are e-related transcendental numbers.

The starters for the numerical iterative methods are obtained by

graphical methods. See the graphs.

graph{x-ln(y+(y^2-1)^0.5)+1+1/y=0[-0.1 0.1 2.2 2.3]}

graph{x-ln(y+(y^2+1)^0.5)+1+1/y=0[-0.1 0.1 2 2.1]}

graph{x-ln y+1+1/y=0[-.1 .1 3.55 3.65]}