The magnitude of measurement error (in micrometers) for a laser range-finder aimed at…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The magnitude of measurement error (in micrometers) for a laser range-finder aimed at a wall is modeled by
E(d)=de−0.05dcos(0.1d),0≤d≤60,
where d is the distance in centimeters from the sensor to the wall. A calculator is permitted for numerical work.
(a) Evaluate E(0) and E(60). Discuss whether either endpoint can maximize |E| within the closed interval.
(b) Derive E′(d) and factor it completely.
(c) Explain why the equation E′(d)=0 cannot be solved algebraically, and use a numerical root-finding method to approximate all critical points in 0<d<60 to two decimal places.
(d) Determine the value of d in 0<d<60 that maximizes the absolute error |E(d)|. Provide supporting reasoning that includes derivative sign analysis or a suitable second-derivative argument.
(e) State the maximum measurement error, rounded to the nearest 0.01 micrometer, and interpret its practical significance for the device’s calibration.