Math1011_Week08 - Math1011 Section 3.2 Use the definition...

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Math1011 Section 3.2 10/11/2010 Use the definition of the derivative to determine if the following function differentiable at x = 0? f ( x ) = x
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Math1011 Section 3.2 1 Use the definition of the derivative to determine if the following function differentiable at x = 0? f ( x ) = x 0 0 0 0 Let's use the definition of the derivative at 0 ( ) ( ) '( ) |0 | |0| | | '(0) Let's look at the limit from the left, and the limit from the right | | from the left, n lim lim lim lim h h h h f x h f x f x h h h f h h h h + = + = = 0 0 0 umbers are negative so |h|= - h 1 | | from the right, numbers are positive so |h|= h 1 Since the limit from the left differs from the limit from the right the limit does not lim lim lim h h h h h h h h h + + = − = | | h 0 exist, so f'(0) = lim = DNE Thus, f (x) is NOT differentiable at x = 0 h h (Doc #1011w.32.02)
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Math1011 Section 3.2 2 Use the definition of the derivative to determine if the following function differentiable at x = 0? 0 ( ) 1 0 x x f x x x = + >
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Math1011 Section 3.2 2 Use the definition of the derivative to determine if the following function differentiable at x = 0? 0 ( ) 1 0 x x f x x x = + > 0 0 Let's use the definition of the derivative ( ) ( ) '( ) Let's look at the limit from the left, and the limit from the right From the left (0 ) 0 so use ( ) (thus (0 ) ) (0 ) ( lim lim h h f x h f x f x h h f x x f h h f h f + = + < = + = + 0 0 0 0 0 0 0 0) (0) 0 = 1 From the right (0 ) 0 so use ( ) 1 (thus (0 ) 1) (0 ) (0) ( 1) (0) 1 0 1 = Since the limit from the left di lim lim lim lim lim lim lim h h h h h h h h f h h h h h h h f x x f h h f h f h f h h h h h + + + + = = = + > = + + = + + + + + h = = = ffers from the limit from the right the limit does not exist, so f'(0) DNE Thus, f (x) is NOT differentiable at x = 0 (Doc #1011w.32.03.doc)
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Math1011 Section 3.2 3 Given the graph of ƒ, sketch ƒ ‘
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Math1011 Section 3.2 3 Given the graph of ƒ, sketch ƒ ‘ (Doc #1011w.32.03)
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