From: "Saved by Windows Internet Explorer 7" Subject: Continuity and Differentiability Date: Fri, 13 Nov 2009 14:35:38 -0700 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_000B_01CA646E.91DBED10" X-MimeOLE: Produced By Microsoft MimeOLE V6.0.6002.18005 This is a multi-part message in MIME format. ------=_NextPart_000_000B_01CA646E.91DBED10 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://people.hofstra.edu/Stefan_Waner/realworld/calctopic1/contanddiffb.html Continuity and Differentiability

Continuity & Differentiability=20
miscellaneous on-line topics for
Calculus Applied to the Real World =
Part B:=20 Differentiability=20

              Return to Main Page
Part A:=20 Continuity
Exercises for=20 This Topic
= Index of=20 On-Line Topics
Everything for=20 Calculus
= Everything for=20 Finite Math
<= FONT=20 face=3D"arial, helvetica, sans-serif" color=3Dblue = size=3D2>Everything for=20 Finite Math & Calculus
Utility:=20 Function Evaluator & Grapher=20

Part B: = Differentiability=20

Note To understand this topic, you will need to be familiar = with=20 derivatives and limits, as discussed in the chapter on the subject in = Calculus Applied to the Real World. If you = like, you=20 can review the topic=20 summary material on derivatives and limits or, for a more detailed = study,=20 the on-line=20 tutorials on derivatives and limits.=20

To begin, we recall the definition of the derivative of a function, = and what=20 it means for a function to be differentiable.=20

Derivative; = Differentiability=20
The derivative of the function f at the point a in its = domain=20 is given by=20
    f'(a) =3D lim
    h0
    f(a+h) - f(a)

    h

We say that function f is differentiable at the point a in = its=20 domain if f'(a) exists.=20

Differentiable on a Subset of the=20 Domain
The function f is differentiable on the = subset S=20 of its domain if it differentiable at each point of S.=20

Note=20
A function can fail to be differentiable at a point a if=20 either lim
h0
f(a+h) - f(a)

h
 does not exist, or is = infinite.
In the former=20 case, we sometimes have a cusp on the graph, and in the latter = case, we=20 get a point of vertical tangency.=20


Example 1 Functions Not = Differentiable at=20 Isolated Points

Determine points of non-differentiability of the following = functions

Solution

=20

(a) The power rule tells us that f(x) =3D (x-1)1/3 has derivative f'(x) =3D (1/3)(x-1)-2/3 everywhere where this expression is = defined, and=20 is not diffeentiable when (1/3)(x-1)-2/3 is not defined. Since (x-1) has a negative exponent, f'(x)=20 is not defined when x =3D 1, and so f is not differentible there. In = fact, direct=20 calculation shows that=20

showing=20 that f is not differentiable at x =3D 1.=20

=20
(b) Since g(x) =3D |x+2| =3D -(x+2)
 
x+2 		  if x = -2
 
if x > -2		   ,
and since we know that both = -(x+2) and x+2 are differentiable, the only point = where=20 something can go wrong is when x =3D -2. At = this point,=20 we can compute the limit of the difference quotient directly:=20

However, this limit does not = exist (see=20 Example 2 in Section 6 of the chapter on derivatives in Calculus Applied to the Real World) since the = left- and=20 right limits differ. =20

(c) The quotient rule tells us that r(x) =3D x2/(x = - 1) is differentiable everywhere except at x =3D = 1. However,=20 x =3D 1 is not in the domain of r, and so r is differentiable at every = point of=20 its domain.=20

As we see in the graph on the right, there are no points of vertical = tangency=20 or cusps.=20

Before We Go On... =

As you can=20 see, the graphs provide immediate information as to where to look for a = point of=20 non-differentiability: a point where there appears to be a cusp or a = vertical=20 tangent.=20

Here is one for you.=20

Example 2 Points of = Non-Dfferentiability=20

    Q

    		f(x) =3D (x-1)4/3 = is =20 at x =3D 1 =

    Q

    		f(x) =3D (x-1)2/3 = is =20 at x =3D 1 =

    Q

    		f(x) =3D (x-1)-1/3 is =20 at x =3D 1 =

    Q

    		f(x) =3D |x-1|4/3 = is =20 at x =3D 1 =20

Q In Part=20 A we discussed continuity, and here we discussed differentiability. = Are all=20 continuous functions differentiable? Are all differentiable functions=20 continuous?
A Briefly: =
(a) Not=20 all continuous functions are differentiable. For instacne, the = closed-form=20 function f(x) =3D |x| is continuous at every real number (including x = =3D 0), but=20 not differentiable at x =3D 0.
(b) However, every differentiable = function is=20 continuous. More precisely, we have the following theorem.=20

Theorem Differentiability Implies=20 Continuity=20

If f is differentiable at a, then it is continuous at a.=20

Proof
Suppose that f = is=20 differentiable at the point x =3D a. Then we know that=20

Thus,=20 This gives=20 If we take x to be = a+h, then h =3D=20 x-a, and the above result can be written as=20 In other words,=20 which means that f is = continuous at x =3D=20 a. You can now either go on and try the rest of the exericses in the exercise=20 set for this topic.
 

Return to Main Page
Part A:=20 Continuity
Exercises for=20 This Topic
= Index of=20 On-Line Topics
Everything for=20 Calculus
= Everything for=20 Finite Math
<= FONT=20 face=3D"arial, helvetica, sans-serif" color=3Dblue = size=3D2>Everything for=20 Finite Math & Calculus
Utility:=20 Function Evaluator & Grapher=20

Last Updated:October, 1999
Copyright =A9 1999 StefanWaner and Steven R. Costenoble=20
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