Differentiability and Continuity

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Presentation transcript:

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

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

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…

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!)

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.

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

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.

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