At a certain industrial site

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

At a certain industrial site, the instantaneous rate (in milligrams per minute) at which a pollutant enters a reservoir is modeled for

0\le t\le 12

f(t)=3t^{2}e^{-0.2t}.

A monitoring program computes the accumulated pollutant load according to the function

P(x)=\int_{h(x)}^{12} f(t)\,dt,\quad \text{where } h(x)=\ln\bigl(1+x^{2}\bigr),\; 0\le x\le 2.

The variable

x

represents a tunable system parameter (dimensionless). (a) Derive an explicit formula for

P'(x)

in terms of

x

. (b) Evaluate

P'(1)

and interpret its sign in the context of the monitoring program. (c) Determine all

x

in the interval

[0,2]

for which

P'(x)=0

. Explain what each such value signifies about the sensitivity of the accumulated pollutant load to changes in the parameter

x

. Use analytical or numerical methods as appropriate, and support your reasoning.