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)=(A of x)=∫2g(x)f(t)dt.
Answer the following questions without performing any numerical calculations.
a. Part 1: (a) State precisely the hypotheses that guarantee A is differentiable at a point x=c in its domain.
b. 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.
c. Part 3: (c) Suppose f(t)≥0 on [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.