Consider the function defined by f(x)= cases x^2, & x<1, 2x-1, & x 1

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

Problem

Consider the function ff defined by f(x)={x2,x<1,2x−1,x≥1.f(x)=\begin{cases} x^2, & x<1,\\ 2x-1, & x\ge1. \end{cases} Respond to the prompts below regarding differentiability at x=1x=1. (a) Verify that ff is continuous at x=1x=1. (b) Determine lim⁡x→1−f(x)−f(1)x−1\displaystyle\lim_{x\to1^-}\frac{f(x)-f(1)}{x-1}. (c) Determine lim⁡x→1+f(x)−f(1)x−1\displaystyle\lim_{x\to1^+}\frac{f(x)-f(1)}{x-1}. (d) Decide whether ff is differentiable at x=1x=1. Support your conclusion with the computations from parts (b) and (c).

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