1. Differentiability and Continuity
AP Calculus
September 22, 2016
Mrs. Agnew

2. Essential Question Essential Vocabulary
What is the relationship between continuity and differentiability?
Essential Vocabulary
Numerical Derivative
Non-differentiability
Continuity

3. Non-Differentiability
When is a function not differentiable?
Remember: derivative = slope of tangent
Can you sketch the graph of a function that has a point of nondifferentiability?
Other examples and calculator lies…

4. THE BIG SIX There are 6 BIG THEOREMS OF CALCULUS.
When you “know” a theorem, you can…
State It
Draw It
Apply It
Prove It
Love It (You can’t live without it!)

5. BIG THEOREM #1
IF A FUNCTION f(x) IS DIFFERENTIABLE AT x = a, THEN IT IS CONTINUOUS AT x = a.
Is converse of the theorem true??? Explain and give examples.

6. Differentiability & Continuity
What is the contrapositive of CALC’S BIG THEOREM #1?
Does differentiability imply continuity?
Does continuity imply differentiability?

7. Nondifferentiability
Where is a function not differentiable?
Jumps, gaps, or holes (not continuous)
Kinks/corner
Vertical tangent lines
Piecewise functions are often “kinky” at the border points.

8. Practice and Homework
Using the graph of a function, identify points of nondifferentiability and state why the function is not differentiable at those points.
Homework: page 106
#83 – 88, 93 – 98, 101 – 104