Consider the piecewise–defined function
f(x)={x2+1,3x−1,x<1,x≥1.
The function is continuous at x=1. Your task is to investigate whether it has an instantaneous rate of change at that point.
a. Part 1: (a) Compute h→0−limhf(1+h)−f(1). Show all algebraic steps.
b. Part 2: (b) Compute h→0+limhf(1+h)−f(1). Show all algebraic steps.
c. Part 3: (c) Determine whether the instantaneous rate of change of f exists at x=1. Justify your answer by comparing the limits obtained in parts (a) and (b).
d. Part 4: (d) Describe geometrically what your conclusion in part (c) implies about the graph of f at x=1. Discuss both the slope of the tangent line and the behavior of the function on either side of the point.