A function is defined on the entire real line and is known to be continuous everywhere
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A function f is defined on the entire real line and is known to be continuous everywhere. Its first derivative f′ is given analytically by
f′(x)=⎩⎨⎧3x+4,x+1,ln(x−1),x<−1,−1≤x<2,x>2.
No additional information about f is provided. Assume all expressions are valid on their stated domains.
(1) Part 1: (a) Evaluate the left–hand and right–hand limits of f′(x) as x approaches −1 and as x approaches 2. State whether f′ can be made continuous at each of these x–values by an appropriate definition of f′(x) at the point.
(2) Part 2: (b) Using your work from part (a), determine all x–values at which f fails to be differentiable. Justify your answer by referencing the definition of differentiability in terms of the derivative.
(3) Part 3: (c) For each x–value identified in part (b), classify the type of non-differentiability exhibited by the graph of f (for example, sharp corner, cusp, vertical tangent, or discontinuity of f). Provide reasoning based on the behavior of f′ near the point.