1. Objectives: 1.Be able to make a connection between a differentiablity and continuity. 2.Be able to use the alternative form of the derivative to determine if the derivative exists at a specific point. Critical Vocabulary: Slope, Tangent Line, Derivative Warm Ups: 1. Find the slope of the tangent line to the graph of f(x) = x 3 + 2x at the point (1, 3). 2. Find the slope of the tangent line to the graph of f(x) = x 2 – 3x -2 at the point (2, 13/4).

2. Warm Ups: 1. Find the slope of the tangent line to the graph of f(x) = x 3 + 2x at the point (1, 3).

3. Warm Ups: 2. Find the slope of the tangent line to the graph of f(x) = x 2 – 3x -2 at the point (2, 13/4).

4. I. Differentiability and Continuity If a function is NOT __________ at a certain point, (say, x = c) then it is also not _________________ at x = c. Greatest Integer Function: f(x)=[[x]] Let’s look at when x = 0 We notice the graph is not continuous at x = 0 because we have a gap. We can’t take a ___________ at a gap We can show this algebraically by using an alternative form of the limit definition of the derivative. This requires that the one-sided limits exist and are equal

5. I. Differentiability and Continuity x f(x) -.5-.1-.01.5.1.010

6. I. Differentiability and Continuity Example: A graph that contains a sharp turn Since the limits are ______, we can conclude that the function is not differentiable at ______ and no tangent line exists at _____.

7. I. Differentiability and Continuity Example: A graph that contains a Vertical Tangent Line Since the limit is ____________, we can conclude that the tangent line is ____________ at x = 0.

8. 1.If a function is ____________ (you can take the derivative) at x = c, then it is ________ at x = c. So, ______________ implies ______________. 2.It is possible for a function to be __________ at x = c and ______ be differentiable at x = c. So, _______ does not imply _____________ (Sharp turns in graphs and vertical tangents). I. Differentiability and Continuity

9. II. Differentiability and Continuity Example: Use the alternative form of the derivative to find the derivative at x = c. f(x) = x 3 + 2x, c = 1

10. II. Differentiability and Continuity Example: Describe the x-values at which f is differentiable

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