Section 2.6Differentiability

Local Linearity• Local linearity is the idea that if we look at any

point on a smooth curve closely enough, it will look like a straight line

• Thus the slope of the curve at that point is the same as the slope of the tangent line at that point

• Let’s take a look at this idea graphically

3 2( ) 9 6f x x x x

Once we have zoomed in enough, the graph looks linear!

Thus we can represent the slope of the curve at that point with a tangent line!

The tangent line and the curve are almost identical!

Let’s zoom back out

Differentiability• We need that local linearity to be able to

calculate the instantaneous rate of change– When we can, we say the function is differentiable

• Let’s take a look at places where a function is not differentiable

• Consider the graph of f(x) = |x|

• Is it continuous at x = 0?• Is it differentiable at x = 0?

– Let’s zoom in at 0

• No matter how close we zoom in, the graph never looks linear at x = 0– Therefore there is no tangent line there so it is not

differentiable at x = 0

• We can also demonstrate this using the difference quotient

h

xfhxfxf

h

)()(lim)('

0

Definition

• The function f is differentiable at x if

exists

• Thus the graph of f has a non-vertical tangent line at x

• We have 3 major cases– The function is not continuous at the point– The graph has a sharp corner at the point– The graph has a vertical tangent

h

xfhxfxf

h

)()(lim)('

0

Example

• Note: This is a graph of • It has a vertical tangent at x = 0

– Let’s see why it is not differentiable at 0 using our power rule

Example

31

)( xxf

• Is the following function differentiable everywhere?

• Graph• What values of a and b make the following function

continuous and differentiable everywhere?

Example

0for

0for)(

2 xx

xxxf

0for)1(

0for2)(

2 xxb

xaxxg

  13) A cable is made of an insulating material in the shape of a long, thin cylinder of radius r0. It has electric charge distributed evenly throughout it. The electric field, E, at a distance r from the center of the cable is given by

• Is E continuous at r = r0?

• Is E differentiable at r = r0?

• Sketch a graph of E as a function of r.

0

20

0

kr for r r

E rk for r rr