Thoughts The function F(x) has the following graph. Match the following Values. 1)F’(-3) = 2)F’(-2) = 3)F’(0) = (a) 1 (b) -2 (c) 0.

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Presentation transcript:

Thoughts The function F(x) has the following graph. Match the following Values. 1)F’(-3) = 2)F’(-2) = 3)F’(0) = (a) 1 (b) -2 (c) 0

AP Calculus Unit 2 Day 3 Derivatives of Graphs and Differentiability

Derivatives we know Y = x+1

Derivatives we know Y = x 2

Derivatives we know

Y = |x|

Differentiable An adjective – Describes a noun, which in this case is a function Definition: – Capable of being differentiated If a function is differentiable then it is possible to find the derivative.

FIRST and FOREMOST To have differentiability at a given x-value the function MUST be continuous at that x-value Example

Example #1—Does exist? First is the function continuous at x=0? Justify your answer. Why is the discontinuity at x=0 a problem?

Discontinuity  Nondifferentiability

SUPER SIZED Conclusion Discontinuity implies non-differentiability In other words, a function must be continuous at x=a in order to be differentiable at x=a

A helpful shortcut with some useful notation

So.... Does this mean that continuity implies differentiability? Let’s explore with this example:

Does Continuity Mean Differentiability? Is the function continuous at x=-2 ? Justify your answer. Now, lets explore the differentiability at x=-2.

NOTE: Absolute value functions are NEVER differentiable at the vertex (“corner point”). Likewise, functions that have a “cusp” like are never differentiable at the cusp. However, you can not use these reasons as justifications for non-differentiability. You must resort to using the limit definition of a derivative to prove non-differentiability

Summarize 1.Confirm continuity 2.Find derivative of the “pieces” 3.Compare the value of the left and right derivatives: Left and right derivatives equal then the derivative exist. Left and right derivatives not equal then the derivative does not exist.

Another Practice Problem 1. Confirm continuity Is f(x) differentiable at x=2 2. Write derivative function 3. State left and right hand derivatives and make a conclusion.

CHECKING IN! Left Derivative ≠ Right Derivative Conclusion: NOT Differentiable at x=a Discontinuous at x=a OR

CHECKING IN! Conclusions: Continuous at x=a Left Derivative = Right Derivative Differentiable at x=a