For each positive constant

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

k

, define the curve

C_k

x(t)=t,\; y(t)=k e^{t}

0\le t\le 1

. Investigate how the total length of

C_k

depends on the parameter

k

. (a) Write the speed

v_k(t)

C_k

as a function of

t

and

k

. Simplify. (b) Show that for every fixed

t

[0,1]

, the speed

v_k(t)

is an increasing function of

k

. Provide derivative-based or algebraic justification. (c) Explain why the total length

S(k)

C_k

[0,1]

must also be an increasing function of

k

. Give a rigorous argument that does not evaluate any definite integrals. (d) Determine whether the ratio

\dfrac{S(k)}{k}

is increasing, decreasing, or neither as

k

increases. Support your conclusion with appropriate reasoning rooted in your earlier work.