Mathematics

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]

: I.

C_{1}: y=p(x)

II.

C_{2}: y=2p(x)

III.

C_{3}: y=p(2x)

for

0\le x\le \tfrac12

(so that each curve begins at

x=0

and ends when its

x

–coordinate reaches

1

). Assume

p'(x)>0

for all

x

(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.