Let for all real numbers

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

Problem

Let

f(x)=\sin x + 2

for all real numbers

x

. Define the accumulation function

A(t)=\int_{\pi/2}^{t}f(x)\,dx

. (a) Calculate

A(\pi/2)

. (b) Determine

A'(t)

and interpret its meaning in terms of

f

. (c) Evaluate

A(\pi)

. (d) Find all values of

t

in the interval

[\pi/2,\,3\pi/2]

at which

A

has a relative maximum or minimum, and classify each critical point with justification.