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 STEP 3 (c) Is f continuous at a Is f differentiable at a Now we need to verify whether the function is continuous at a=2 For this we have to show lim f(x) = lim f(x) x2 + x2 lim +(x) = lim |x+2| = lim x2 =+0 x2 x2 x2 lim f(x) = lim |x2| = lim (x2) = 0 x2 x2 x2 Thus the condition is satisfied. Thus the function f(x) is continuous at a=2. It is already known that the derivative f(a) exists if and only if . f+(a) = f a) Since ,in this case f (a) = / f (a) the derivative does not exist. + Therefore f(x) = |x2| is not differentiable at a=2. .