dy/dx=sinh(x) A tangent line through the origin has equation y=mx. If it meets the graph at x=a, then ma=cosh(a) and m=sinh(a). Therefore, asinh(a)=cosh(a). Use Newton's Method to solve for a

Maverick Avery

Maverick Avery

Answered question

2022-10-23

d y / d x = sinh ( x ) A tangent line through the origin has equation y = m x. If it meets the graph at x = a, then m a = cosh ( a ) and m = sinh ( a ). Therefore, a sinh ( a ) = cosh ( a ).
Use Newton's Method to solve for a

Answer & Explanation

Pradellalo

Pradellalo

Beginner2022-10-24Added 16 answers

You are looking for the zero of function
f ( a ) = a sinh ( a ) cosh ( a )
for which
f ( a ) = a cosh ( a ) and f ( a ) = a sinh ( a ) + cosh ( a )
Since the function is even, you have two symmetric solutions.
What you can notice is that f ( 0 ) = 1, f ( 0 ) = 0 and f ( 0 ) = 1. So, to generate a guess, perform a Taylor expansion around a = 0; this would give
f ( a ) = 1 + a 2 2 + a 4 8 + O ( a 6 )
which is a quadratic in a 2 . So, an estimate is
a 0 = 2 ( 3 1 )
Now, use Newton method which generates as itegrates
a n + 1 = a n f ( a n ) f ( a n ) = a n + 1 a n tanh ( a n )
which should converge very fast.

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