A reservoir receives water at a rate modeled by the continuous function

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

Problem

r(t)=4+\sin(2t)

, measured in cubic meters per hour, where

t

is the number of hours after midnight. For a certain environmental study, the net volume of water accumulated in the reservoir between the time

t=g(x)

and the time

t=h(x)

is given by

V(x)=\int_{g(x)}^{h(x)} r(t)\,dt,

where

g(x)=x

and

h(x)=x^2+3

for

0\le x\le 4

. (a) Obtain an expression for

V'(x)

in terms of

x

. (b) Solve the equation

V'(x)=0

for

0\le x\le 4

. Interpret each solution in terms of the study. (c) Compute

V''(x)

and determine whether each critical point found in part (b) corresponds to a local maximum or local minimum of

V

. Provide a justification without evaluating

V

itself. (d) Explain what

V'(2)

represents in this context, and determine its numerical value, including appropriate units.