Let the function be defined piecewise by g(x)= cases -2x+1, & x<0, 2x+1, & x 0

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

Problem

Let the function gg be defined piecewise by g(x)={−2x+1,x<0,2x+1,x≥0.g(x)=\begin{cases} -2x+1, & x<0,\\ 2x+1, & x\ge 0. \end{cases} Answer the following questions about the behavior of gg at x=0x=0. (a) Evaluate g(0)g(0) and determine lim⁡x→0g(x)\displaystyle\lim_{x\to0}g(x). Explain what your results imply about the continuity of gg at x=0x=0. (b) Compute lim⁡h→0−g(0+h)−g(0)h\displaystyle\lim_{h\to0^-}\frac{g(0+h)-g(0)}{h}. Show all algebraic steps. (c) Compute lim⁡h→0+g(0+h)−g(0)h\displaystyle\lim_{h\to0^+}\frac{g(0+h)-g(0)}{h}. Show all algebraic steps. (d) Using your answers to parts (b) and (c), determine whether gg possesses an instantaneous rate of change at x=0x=0. Provide a clear justification that references the limits you calculated.

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