The piecewise function h is defined as follows: h(x) = cases 5x+2, & x < 3
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The piecewise function h is defined as follows:
h(x)={5x+2,x2−4x+k,x<3,x≥3,
where k is a real constant.
Answer the questions that follow.
(a) Determine the value of k for which h is differentiable at x = 3. Justify your answer by evaluating appropriate limits.
(b) For the k found in part (a), compute h′(3). Show all limit manipulations.
(c) Describe geometrically how your work in parts (a) and (b) ensures that the two pieces of h connect to form a single smooth curve at x = 3.