MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
For a fixed real number k>0, consider the family of functions
fk(x)=∣x∣k.
(1) Part (a) — General form of the difference quotient for x=0
(2) Part (b) — Limit evaluation at the origin
(3) Part (c) — Parameter values ensuring differentiability at 0
(4) Part (d) — Value of the derivative and interpretation of smoothness