For a fixed positive constant

MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

k

, consider the curve

y=\dfrac{1}{k}\cosh(kx)

on the interval

0\le x\le b

, where

b>0

. (a) Show that

\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\cosh(kx)

. Present a clear chain of equalities that leads to your conclusion. (b) Derive a closed‐form expression for the arc length

L(k,b)

of the curve on

[0,b]

in terms of

k

and

b

. Express your final answer without an integral symbol. (c) Evaluate

L(2,\ln 3)

exactly, where

k=2

and

b=\ln 3

. Show all intermediate steps. (d) Determine

\displaystyle \lim_{k\to 0^{+}} L(k,b)

for a fixed

b>0

and interpret the limiting value in the context of the family of curves

y=\dfrac{1}{k}\cosh(kx)