MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Let a function f be continuous on the interval [−3,5]. The graph of f consists of straight-line segments joining the points (−3,0), (−1,2), (2,−1), and (5,3).
An accumulation function g is defined by
g(x)=∫−3xf(t)dt.
Assume areas below the t-axis are treated as negative.
(a) Determine all intervals on (−3,5) where g is increasing and all intervals where g is decreasing. Justify your answer.
(b) Identify the x-coordinates of all critical points of g on (−3,5). For each critical point, classify it as the location of a relative maximum, a relative minimum, or neither. Provide a reason for each classification.
(c) Describe the concavity of g on each subinterval determined by the given breakpoints of f. Explain how the shape of the graph of f supports your conclusion.
(d) Compute g(5). Show all steps, including any geometric interpretation you use.