MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A differentiable function g is defined on [0,7]. The graph of y=g′(x) consists of three straight-line segments joining the points (0,0), (2,2), (5,−2), and (7,0). Additional information is given.
• The signed area between the graph of g′ and the x-axis is 3 on [0,2], −6 on [2,5], and 1 on [5,7] (units are square units).
• g(0)=3.
(a) Calculate the values of g(2), g(5), and g(7). Show your work.
(b) Determine the x-coordinate(s) at which g attains its absolute maximum and its absolute minimum on the interval [0,7]. State the corresponding function values.
(c) Justify your conclusions in part (b) by relating the accumulated change in g to the behavior of g′.