Chapter 3.2, Problem 72E

One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by

$f_{+}{}^{\prime }(a)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f(a)}{h}and{f}_{-}{}^{\prime }(a)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f(a)}{h},$

respectively, provided these limits exist. The derivative f′(a) exists if and only if f₊′(a) = f₋′(a).

1. a. Sketch the following functions.
2. b. Compute f₊′(a) and f₋′(a) at the given point a.
3. c. Is f continuous at a? Is f differentiable at a?

32. $f(x)=\{\begin{array}{ll}4-x^2\hfill & \text{if}x\le 1\hfill \\ 2x+1\hfill & \text{if}x>1\hfill \end{array};a=1$