Chapter 32 How Might Fail to Exist A function will not have a derivative at a point where the slopes of the secant lines fail to approach a limit as Some of the common ways where a function fails to have a derivative ID: 264690
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Differentiability
Chapter 3.2Slide2
How
Might Fail to Exist
A function will not have a derivative at a point
where the slopes of the secant lines fail to approach a limit as Some of the common ways where a function fails to have a derivative:A corner, where the one-sided derivatives differA cusp, where the slopes of the secant lines approach from one side and from the otherA vertical tangent, where the slopes of the secant lines approach either or from both sidesA discontinuity, where one or both of the one-sided derivatives will not exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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How
Might Fail to Exist
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Example 1: Finding Where a Function is Not Differentiable
Find all points in the domain of
where
is not differentiable.
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Example 1: Finding Where a Function is Not Differentiable
Since this is an absolute value function, it will have a “corner”. We can see where this is by rewriting
as a piecewise function:
If we use the alternative definition of derivative, then
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Example 1: Finding Where a Function is Not Differentiable
From the left we have
Since the left- and right-hand limits are different, then the derivative does not exist at
.
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Differentiability Implies Local Linearity
An important notion about differentiability that will show up later is that differentiable functions are
locally linearThis means that, if we view the graph of a function at a point
over a sufficiently small interval of containing
, the graph will appear to be linear, or nearly linearThat is, the graph will appear to match its tangent line at that point 13Slide14
Differentiability Implies Local Linearity
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Differentiability Implies Continuity
Note the definition of the derivative did not require that the function be continuous, though many of the theorems you will see later do require this condition
The reason is that differentiability implies continuity
Another way to say this is that, if a function is differentiable at some
, then must be continuous at Or using the contrapositive, if is discontinuous at , then it is not differentiable at This idea (which we will prove) comes up often in AP exam questions when continuity is a required condition 15Slide16
Differentiability Implies Continuity
THEOREM:
If
has a derivative at
, then is continuous at .Recall that the three requirements for a function to be continuous at a point are: must be defined must exist
Note that 3 includes 1 and 2 (if it’s true). So by proving that
, we will have proven continuity at
.
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Differentiability Implies Continuity
PROOF
We will use the alternate limit definition.
By the Limit Product Property
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Differentiability Implies Continuity
PROOF
Since
, then by the Limit Difference Property
Now we can say that, if a function is differentiable at some number
of its domain, then it is automatically continuous at
. Or,
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Differentiability Implies Continuity
Is the converse of this theorem also true?
That is, does continuity imply differentiability?As you saw earlier, a function may be everywhere continuous, but may also be non-differentiable at one or more pointsThe first three of the four examples from the beginning of this lesson are of everywhere continuous functions that were all non-differentiable at
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Exercise 3.2
Online exercise
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