A function is represented by a polygonal (piecewise-linear) curve consisting of three…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A function f is represented by a polygonal (piecewise-linear) curve consisting of three straight line segments that join the ordered pairs (−5,4), (−2,1), (1,4), and (4,0) in that order.
Answer the following questions about the derivative of f.
(a) For each open sub-interval on which f is linear, write an explicit formula for f′(x). Express your answer as a piecewise-defined function on the domain (−5,4).
(b) State all x–values in (−5,4) at which f′(x) does not exist and explain the geometric reason for each nondifferentiability.
(c) Sketch the complete graph of y=f′(x) on [−5,4], including any open or closed dots needed to show discontinuities. Clearly label the numerical value of the derivative on each interval where it is constant.
(d) Determine whether the derivative function f′ is continuous on (−5,4). Justify your conclusion using appropriate limit language.