• 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

f x
 x
3  x
2 
9 x

6

Thus we can represent the slope of the curve at that point with a tangent line!
Once we have zoomed in enough, the graph looks linear!

Let’s zoom back out
The tangent line and the curve are almost identical!

• 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 f ' ( x )
 h lim

0 f ( x
 h )
 f ( x ) h

Definition
• The function f is differentiable at x if f ' ( x )
 h lim

0 f ( x
 h )
 f ( x ) h 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

Example

Example
• Note: This is a graph of f ( x )
 x
1
3
• It has a vertical tangent at x = 0
– Let’s see why it is not differentiable at 0 using our power rule

Example
• Is the following function differentiable everywhere?
•
Graph f ( x )



 x x
2 for for x x

0

0
• What values of a and b make the following function continuous and differentiable everywhere?
g ( x )

 ax

 b ( x


2
1 )
2 for x

0 for x

0

13) A cable is made of an insulating material in the shape of a long, thin cylinder of radius r
0
. 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 kr
E
 
 k r
0
2 r
• Is
E continuous at r = r
0
?
for r
 r
0 for r
 r
0
• Is
E differentiable at r = r
0
?
• Sketch a graph of
E as a function of r .