A smooth racetrack is modeled by the graph of a differentiable function on
MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A smooth racetrack is modeled by the graph of a differentiable function y=w(x) on 0≤x≤2. The track begins at (0,0) and ends at (2,0). Engineers are considering three design modifications, each of which transforms the function w as described below.
Design A: Reflect the graph of w across the y–axis and then translate it right by 2 units.
Design B: Translate the graph of w up by 1 unit.
Design C: Stretch the graph of w vertically by a factor of 23.
Assume w′(x) is never zero on (0,2).
(a) For each design, state whether the resulting curve has the same arc length, a greater arc length, or a smaller arc length than the original curve. Provide a brief qualitative explanation in each case.
(b) Rank the arc lengths of the three modified curves and the original curve from smallest to largest, clearly indicating any equalities and defending your ordering with appropriate reasoning.