MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A family of functions is defined by fk(x)=tan(kx), where k is a non-zero real constant.
(a) Describe how the parameter k affects the steepness of the graph of fk near the origin. Support your answer by referencing the derivative of fk.
(b) Show that the slope of the graph of fk at x=0 is k and interpret this result in the context of your description from part (a).
(c) Explain why both fk and its derivative are periodic with period π/∣k∣, and discuss how this period relates to the spacing of vertical asymptotes of fk.