Differentiability

Piecewise functions may or may not be differentiable on

their domains.

To be differentiable at a point cx = , the function must be

continuous, and we will then see if it is differentiable.

Let’s consider some piecewise functions first.

Let ( )0,

0,

>

≤−=

xx

xxxf

First we will check to prove continuity at 0=x

Now we will consider differentiability at 0=x

Now let’s try:

1,54

1,38)(

2>+

≤−=

xx

xxxg

Is )(xg continuous AND differentiable at ?1=x

Is )(xh continuous and differentiable at ?3=x

( )3,12

3,842

>−

≤+−=

xx

xxxxh

Now let’s try something trickier:

If 1,2

1,43)(3

2

>++

≤+=

xcbxx

xxxxf

Find b and c so that f(x) is differentiable at x = 1

Let’s work on continuity first.

Now work on differentiability

Let 2,6

2,10)(

2

2

≥+−

<+=

xbxx

xaxxf

Find the values of a and b such that )(xf is differentiable

Homework: page 118 #113 and #114 [show ALL steps

with GOOD notation]