Consider the piecewise–defined function f(x)= cases x^ 2 +1, & x<1, 3x-1, & x 1

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

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

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