The function is twice differentiable on and satisfies and

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

Problem

The function

p

is twice differentiable on

[0,1]

and satisfies

p(0)=0

and

p(1)=3

. Consider the three curves defined on

[0,1]

: (a) Explain qualitatively how a vertical scaling (multiplying by a constant) affects the quantity

\sqrt{1+(p'(x))^{2}}

. (b) Explain qualitatively how a horizontal compression (replacing

x

2x

) affects the same quantity. (c) Rank the arc lengths of

C_{1}

C_{2}

, and

C_{3}

from smallest to largest. Provide a clear justification for each comparison you make.