Let be continuous on an open interval containing

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

Problem

Let

f

be continuous on an open interval containing

[2,8]

, and let

g

be a differentiable function whose range is contained in

[2,8]

. Define an accumulation function

A(x)=\bigl(A\text{ of }x\bigr)=\int_{2}^{\,g(x)} f(t)\,dt.

Answer the following questions without performing any numerical calculations. (1) Part 1: (a) State precisely the hypotheses that guarantee

A

is differentiable at a point

x=c

in its domain. (2) Part 2: (b) Use appropriate theorems to write a simplified expression for

A'(c)

in terms of

f

g

, and their derivatives evaluated at

c

. Clearly justify each step of your reasoning. (3) Part 3: (c) Suppose

f(t)\ge 0

[2,8]

and

g'(c)<0

. Explain how the sign of

A'(c)

can be determined directly from your expression in part (b), and interpret that sign in the context of accumulated area with a moving upper limit.