The piecewise function is defined for all real numbers.
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The piecewise function p(x)=∣x∣ is defined for all real numbers.
(a) Write the limit expression that represents the derivative of p at x=0 from the right side and an analogous expression from the left side. Do not evaluate yet.
(b) Compute each one-sided limit obtained in part (a).
(c) Using the results from part (b), determine whether p′(0) exists. Justify your conclusion fully, referencing the limits you calculated.
(d) Describe geometrically what your conclusion in part (c) implies about the graph of y=∣x∣ at x=0. Use appropriate terminology related to tangents or slopes.