Transcript Section 2.6 Differentiability

Section 2.6
Differentiability
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
f ( x)  x  x  9 x  6
3
2
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!
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
f ' ( x )  lim
h 0
f ( x  h)  f ( x)
h
Definition
• The function f is differentiable at x if
f ' ( x )  lim
h 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 ) 
• It has a vertical tangent at x = 0
x
1
3
– Let’s see why it is not differentiable at 0 using our
power rule
Example
• Is the following function differentiable everywhere?
x
f ( x)   2
x
for x  0
for x  0
• Graph
• What values of a and b make the following function
continuous and differentiable everywhere?
 ax  2
g (x)  
2
 b ( x  1)
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 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
 kr

E   r2
0
k

 r
for r  r0
for r  r0
• Is E continuous at r = r0?
• Is E differentiable at r = r0?
• Sketch a graph of E as a function of r.