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Section 2.6 Differentiability
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Consider the graph of f(x) = |x|
Is it continuous at x = 0? Is it differentiable at x = 0? Let’s zoom in at 0
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No matter how close we zoom in, the graph never looks linear at x = 0
Therefore there is no tangent line there so it is not differentiable at x = 0 We can also demonstrate this using the difference quotient
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The function f is differentiable at x if exists
Definition The function f is differentiable at x if exists Thus the graph of f has a non-vertical tangent line at x We have 3 major cases The function is not continuous at the point The graph has a sharp corner at the point The graph has a vertical tangent
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Example
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It has a vertical tangent at x = 0
Example Note: This is a graph of It has a vertical tangent at x = 0 Let’s see why it is not differentiable at 0 using our power rule
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Example Is the following function differentiable everywhere? What values of a and b make the following function continuous and differentiable everywhere?
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