1. Differentiability, Local Linearity
Section 3.2a

2. How f (a) Might Fail to Exist
A function will not have a derivative at a point P (a, f(a)) where
the slopes of the secant lines,
fail to approach a limit as x approaches a. Four instances
where this occurs:
1. A corner, where the one-sided derivatives differ.
Example:
There is a corner at x = 0

3. How f (a) Might Fail to Exist
A function will not have a derivative at a point P (a, f(a)) where
the slopes of the secant lines,
fail to approach a limit as x approaches a. Four instances
where this occurs:
2. A cusp, where the slopes of the secant lines approach
infinity from one side and negative infinity from the other.
Example:
There is a cusp at x = 0

4. How f (a) Might Fail to Exist
A function will not have a derivative at a point P (a, f(a)) where
the slopes of the secant lines,
fail to approach a limit as x approaches a. Four instances
where this occurs:
3. A vertical tangent, where the slopes of the secant lines
approach either pos. or neg. infinity from both sides.
Example:
There is a vertical tangent at x = 0

5. How f (a) Might Fail to Exist
A function will not have a derivative at a point P (a, f(a)) where
the slopes of the secant lines,
fail to approach a limit as x approaches a. Four instances
where this occurs:
4. A discontinuity (which will cause one or both of the one-
sided derivatives to be nonexistent).
Example: The Unit Step Function
There is a discontinuity at x = 0

6. Relating Differentiability and Continuity
Theorem: If has a derivative at x = a, then is
continuous at x = a.
Intermediate Value Theorem for Derivatives
If a and b are any two points in an interval on which
is differentiable, then takes on every value
between and
Ex: Does any function have the Unit Step Function as its
derivative?
NO!!! Choose some a < 0 and some b > 0. Then U(a) = –1
and U(b) = 1, but U does not take on any value between –1
and 1  can we see this graphically?

7. Differentiability Implies Local Linearity
Locally linear function – a function that is
differentiable at a closely resembles its own tangent
line very close to a.
Differentiable curves will “straighten out” when we
zoom in on them at a point of differentiability…

8. Differentiability Implies Local Linearity
Is either of these functions differentiable at x = 0?
1. We already know that f is not differentiable at x = 0; its graph
has a corner there. Graph f and zoom in at the point (0,1)
several times. Does the corner show signs of straightening out?
Continued zooming in at the given point (assuming a
square viewing window) always yields a graph with the
exact same shape  there is never any “straightening.”

9. Differentiability Implies Local Linearity
Is either of these functions differentiable at x = 0?
2. Now do the same thing with g. Does the graph of g show signs
of straightening out?
Try starting with the window [–0.0625, ] by
[0.959, 1.041], and then zooming in on the point (0,1).
Such zooming begins to reveal a smooth turning point!!!

10. Differentiability Implies Local Linearity
Is either of these functions differentiable at x = 0?
3. How many zooms does it take before the graph of g looks
exactly like a horizontal line?
After about 4 or 5 zooms from our previous window, the
graph of g looks just like a horizontal line.
 This function has a horizontal tangent at x = 0, meaning
that its derivative is equal to zero at x = 0…

11. How does all of this relate to our topic of local linearity???
Differentiability Implies Local Linearity
Is either of these functions differentiable at x = 0?
4. Now graph f and g together in a standard square viewing
window. They appear to be identical until you start zooming in.
The differentiable function eventually straightens out, while the
nondifferentiable function remains impressively unchanged.
Try the window [– , ] by [0.9795, ]
How does all of this relate to
our topic of local linearity???

12. Guided Practice There is a corner at (2,3), so this function is not
Find all the points in the domain of where
the function is not differentiable.
Think about this problem graphically. We have the graph of the
absolute value function, translated right 2 and up 3:
There is a corner at (2,3),
so this function is not
differentiable at x = 2.

13. Guided Practice
For the given function, compare the right-hand and left-hand
derivatives to show that it is not differentiable at point P.
Left-hand derivative:
Graph the function:
Right-hand derivative:

14. Guided Practice Left-hand derivative: 0 Right-hand derivative: 2
For the given function, compare the right-hand and left-hand
derivatives to show that it is not differentiable at point P.
Left-hand derivative: 0
Graph the function:
Right-hand derivative: 2
Since , the function is
not differentiable at point P.

15. Guided Practice (a) All points in [–2,3] except x = –1, 0, 2
The graph of a function over a closed interval D is given below.
At what points does the function appear to be
(a) differentiable?
(b) continuous but not differentiable?
(c) neither continuous nor differentiable?
(a) All points in [–2,3]
except x = –1, 0, 2
(b) x = –1
(c) x = 0, x = 2

16. Guided Practice The problem is a cusp!!!
The given function fails to be differentiable at x = 0. Tell whether
the problem is a corner, a cusp, a vertical tangent, or a
discontinuity. Support your answer analytically.
Check the one-sided derivatives!!!
The problem is a cusp!!!
(support with a graph???)

17. Guided Practice The problem is a vertical tangent!!!
The given function fails to be differentiable at x = 0. Tell whether
the problem is a corner, a cusp, a vertical tangent, or a
discontinuity. Support your answer analytically.
Check the two-sided derivative!!!
The problem is a vertical tangent!!!
(support with a graph???)