Math 312 Sections 1 2 Lecture Notes FALL 2007 14 Differentiation Definition Let D be a subset of R containing a neighborhood of a that is an open interval a r a r with r 0 We say that a function f D R is differentiable at a if the following limit exists f x f a f 0 a x a x a lim The real number f 0 a is called the derivative of f at a The quotient f x f a x a is referred to as the difference quotient Equivalently setting x a x one gets f a x f a lim f 0 a x 0 x Example 1 Let f x x2 on R Show that f is differentiable at each point a R Solution We have f x f a x 2 a2 x a x a x a Therefore f x f a 2a x a In other words f 0 a 2a for all a in R Example 2 Let f x x on the open interval I 0 Prove that f is differentiable at every point of I Solution For every a I we have lim x a f x f a x a x a x a x a x a 1 x a x a x a x a x a x a It follows f x f a 1 x a x a 2 a lim Thus f 0 a 1 2 a for all a in R Remark Assume that the following right hand and left hand limits exist lim x a f x f a f 0 a x a lim x a f x f a f 0 a x a The function f is differentiable at a if and only if f 0 a and f 0 a exist and are the same Then f 0 a f 0 a f 0 a Example 3 Let f x x defined on R Is f differentiable at zero Solution We have lim x 0 x 1 and x lim x 0 x 1 x Since the right hand and the left hand limits of f at zero are different f is not differentiable at zero Example 4 Let f x x sin x1 for x 6 0 and f 0 0 Prove that f is not differentiable at zero Solution We have x sin x1 1 lim sin lim x 0 x 0 x x The limit does not exist Hence f is not differentiable at zero Local extremum points Definition Let f I R be a function on I where I is an open interval A point c in I is a local maximum of f if f c f x for x c A point c in I is a local minimum of f if f c f x for x c Theorem Let f be a differentiable function on an open interval I If a is a local extremum point then f 0 a 0 Remark The converse of the previous theorem is false For example if f x x3 then f 0 0 0 but a 0 is not a local extremum point