One-sided derivatives? ?The left-hand and right-hand derivatives of a function at a point a are given by ? f(a+h)?f(a) f+(a) = lim h h?0 + and ? f(a+h)?f(a) f?(a) = limh?0 + h provided these limits e?xis? t. The derivative f?(a ? ?? xists if and only if . f+(a) = f (a?? ? a. ?Sketch the following functions. ? ? ? b. ?Compute f (a)an+ f (a)at the ?iven point a. ? c. ?Is f continuous at a? Is f differentiable at a? ? ? ? ? f(?x) = ??x ? 2?; ?a = 2

SOLUTION Given f(x) = |x2|;a = 2 STEP 1 (a). Sketch the following functions f(x) = |x2| STEP 2 (b). Compute f +a)and f ()at the given point a. The left and the right derivatives of the function are f (a) = lim f(a+h)f(a) And f (a) = lim f(a+h)f(a) + h0 + h h0 + h Provided a=2. |2+h2||x2| |h| Therefore f (2+ = lim + h = lim + h h0 h0 At h > 0 , |h| = h |h| h Therefore lim + h = lim +h = 1 h0 h0 Thus we got f (2)+= 1 |2+h2||x2| |h| Then, f (2 = lim h = lim h h0 h0 At h < 0 , |h| =h |h| h Therefore lim h = lim h =1 h0 h0 Thus we got f (2)=1...