Riley hobson bence · Problem 2.49

Preliminary calculus

Mathematics — Methods · arc-length-inequalities

Problem

By noting that

\sinh x < \tfrac{1}{2}e^x < \cosh x

, and that

1 + z^2 < (1+z)^2

for

z > 0

, show that, for

x > 0

, the length

L

of the curve

y = \tfrac{1}{2}e^x

measured from the origin satisfies the inequalities

\sinh x < L < x + \sinh x