MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A function g is defined on the interval −4≤x≤6. The graph of g is described verbally as follows.
• For −4≤x<0, the graph is a straight line segment from (−4,3) to (0,1).
• At x=0 there is a closed (filled) point at (0,1) and an open point at (0,2).
• For 0<x<4, the graph is the right half of a circle with center (0,2) and radius 2 (so the arc joins (0,2) to (4,2) and is concave downward).
• For 4<x≤6, the graph is the curve given by y=3x−4+2.
Answer the following questions about g.
(a) Determine whether g is continuous at x=−4, x=0, x=4, and x=6. For each point, cite the relevant limit or value of g that supports your conclusion.
(b) For each of the same four x–values, decide whether g is differentiable there. Support each answer by referring to appropriate graphical features (such as a corner, cusp, vertical tangent, or discontinuity).
(c) Summarize the relationship you observe between continuity and differentiability for this example.