A car travels along a differentiable road described by for

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

A car travels along a differentiable road described by

y=f(x)

for

x\in[a,b]

. Which reasoning most strongly supports the claim that the distance driven is always at least the horizontal displacement

(b-a)

? (a) Because the elemental distance can be expressed as

ds = dx + |dy|

, integrating gives

s=(b-a)+\int_a^b|dy|\ge b-a

. (b) For each increment

dx>0

, the elemental distance is

ds=\sqrt{1+(dy/dx)^2}\,dx\ge dx

because the square-root term is at least 1; integrating from

a

b

gives

s\ge b-a

. (c) Because

|dy|

is always no smaller than

dx

, accumulating

dy

guarantees a longer path than

dx

alone. (d) The Mean Value Theorem ensures that the average slope on

[a,b]

exceeds

1

, forcing the total length to exceed the horizontal distance.