1. The Derivative (cont.)
3.1

2. A Differential Function is Continuous
If y = f(x) has a derivative at x = c, then f(x) is continuous at x = c.
When the Derivative Fails to Exist
The derivative fails to exist when
The graph of the function has a corner.
The graph of the function has a vertical tangent.
The graph of the function has a break (discontinuity).

3. To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner
cusp
discontinuity
vertical tangent

4. Continuity does not imply Differentiability
Differentiability implies Continuity

5. Find the interval where the function is differentiable.
Limit does not exist, so the function is not continuous and not differentiable at x = 0
Possible point of discontinuity at x = 0

6. Using your calculator Graph |x| + 1 Zoom in on “corner”
Notice the corner does not change
A differentiable curve will “straighten out”

7. Find at x = 2. Example: 1. Graph the function
2. Press 2nd TRACE to enter the CALC Menu
3. Select 6. dy/dx
4. Press 2 and ENTER
5. BE CAREFUL! The calculator gave an answer of The answer is 12!
This should only be used as a check for your homework. You cannot use this method on a test/quiz!!