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MTH 251 – Differential Calculus Chapter 3 – DifferentiationReview – The Derivative at a PointThe Derivative as a FunctionDerivative NotationDerivative at x = a NotationExamples …Slide 7Sketching the Graph of f ’(x) using the Graph of f(x)Slide 9Left & Right Derivatives at a PointSlide 11Where does a derivative NOT exist?Slide 13Slide 14Slide 15Differentiability & ContinuitySlide 17MTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.2The Derivative as a FunctionCopyright © 2010 by Ron Wallace, all rights reserved.Review – The Derivative at a Point•The derivative was defined as the limit of the difference quotient. That is …•If x0 + h = z, then an alternate definition would be …•Note that the result of this limit is a number. That is, the derivative at a specific value of x.( )0 000( ) ( )' limhf x h f xf xh�+ -=Remember: x0 refers to a specific value of x.( )0000( ) ( )' limz xf z f xf xz x�-=-The Derivative as a Function•If we do not specify a specific value of x (i.e. use x instead of x0) we get a function called the derivative of f(x).•That is, the derivative of f(x) is the function …0( ) ( )'( ) limhf x h f xf xh�+ -=( ) ( )'( ) limz xf z f xf xz x�-=-ORf(x+h) xx+hhf(x)f(x+h) – f(x)Derivative Notation•All of the following can be used to designate the function that is the derivative of y = f(x)'f'( )f x'y[ ]( )df xdx[ ]( )xD f xdydxdfdxReminder: The results of these will be a function.Derivative at x = a Notation•All of the following can be used to designate the derivative of y = f(x) at x = a'( )f a[ ]( )x adf xdx=x adydx=[ ]( )xaD f xadfdxReminder: The results of these will be a number.Examples …•Determine the following derivatives …1ddx x������dxdx� �� �ndxdx� �� � IMPORTANT!Memorize these 3 results.Examples …•Determine the following derivatives …71xddx x=������9dxdx� �� �5dxdx� �� �Sketching the Graph of f ’(x) using the Graph of f(x)•Where is the derivative (i.e. slope) zero?•Where is the derivative (i.e. slope) positive?Large or small positive?•Where is the derivative (i.e. slope) negative?Large or small negative?•Where is the derivative (i.e. slope) constant?Function is a line segment.Derivative is a horizontal line segment.Sketching the Graph of f ’(x) using the Graph of f(x)•Example – Sketch the graph of the derivative of the following function.Left & Right Derivatives at a Point•If in the definition of the derivative at a point, you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other.•Right-Hand Derivative at x0•Left-Hand Derivative at x0•If these are equal, then …( )( ) ( )0 000' limhf x h f xf xh++�+ -=( )( ) ( )0 000' limhf x h f xf xh--�+ -=( ) ( )( )00 0' ' 'f x f x f x+ -= =Left & Right Derivatives at a Point•Example: ( )( )( )24' 2 ?' 2 ?f x xff+-= -==Where does a derivative NOT exist?•Cornerleft & right derivatives are different22y x= -Where does a derivative NOT exist?•Corner•Cuspleft & right derivatives are approaching & –21524( 1)y x= + -Where does a derivative NOT exist?•Corner•Cusp•Vertical TangentThe derivative limit is or –31 1y x= + -Where does a derivative NOT exist?•Corner•Cusp•Vertical Tangent•Discontinuitysee the next theorem3 265( 3)x x xyx- -=-Differentiability & Continuity•If f ’(c) exists, then f(x) is continuous at x = c.Proof …Let x c h= +Therefore: lim ( ) ( )x cf x f c�=0lim ( )hf c h�+0lim ( ) [ ( ) ( )]hf c f c h f c�= + + -0[ ( ) ( )]lim ( )hf c h f cf c hh�+ -= + �0 0 0[ ( ) ( )]lim ( ) lim limh h hf c h f cf c hh� � �+ -= + �( ) '( ) 0 ( )f c f c f c= + �= as 0x c h�ޮޮDifferentiability & Continuity•If f ’(c) exists, then f(x) is continuous at x = c.Or … the contrapositive implies …•If f(x) is NOT continuous at x = c, then f ’(c) does not exist.•NOTESIf the derivative does not exist, that does not mean the function is not continuous.If the function is continuous, that does not mean that the derivative exists.Example … the Absolute Value

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