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,

DerivativeWarm Ups: 1. Find the slope of the tangent line to the graph of f(x) = x3 + 2x at the point (1, 3).

2. Find the slope of the tangent line to the graph of f(x) = x2 – 3x-2 at the point (2, 13/4).

Warm Ups: 1. Find the slope of the tangent line to the graph of f(x) = x3 + 2x at the point (1, 3).

Warm Ups: 2. Find the slope of the tangent line to the graph of f(x) = x2 – 3x-2 at the point (2, 13/4).

I. Differentiability and ContinuityIf 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 = 0We 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

I. Differentiability and Continuity

00]][[lim

0

xx

x

00]][[lim

0

xx

x

xf(x)

-.5 -.1 -.01 .5.1.010

cxcfxf

cx

)()(limcxcfxf

cxcfxf

cxcx

)()(lim)()(lim

I. Differentiability and Continuity

Example: A graph that contains a sharp turn 2)( xxf

2)2(2

lim2

x

fxx

2)2(2

lim2

x

fxx

Since the limits are ______, we can conclude that the function is not differentiable at ______ and no tangent line exists at _____.

cxcfxf

cx

)()(limcxcfxf

cxcfxf

cxcx

)()(lim)()(lim

I. Differentiability and Continuity

Example: A graph that contains a Vertical Tangent Line3)( xxf

0)0(lim

3

0

xfx

x

0)0(lim

3

0

xfx

x

Since the limit is ____________, we can conclude that the tangent line is ____________ at x = 0.

cxcfxf

cx

)()(limcxcfxf

cxcfxf

cxcx

)()(lim)()(lim

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

II. Differentiability and Continuity

Example: Use the alternative form of the derivative to find the derivative at x = c.

f(x) = x3 + 2x, c = 1

cxcfxf

cx

)()(limcxcfxf

cxcfxf

cxcx

)()(lim)()(lim

II. Differentiability and ContinuityExample: Describe the x-values at which f is differentiable

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