MTH 251 – Differential Calculus Chapter 3 – DifferentiationZooming in on a FunctionSlide 3Slide 4Slide 5LinearizationPreview of things to come …Slide 8Slide 9Slide 10Slide 11Slide 12DifferentialsSlide 15Slide 16Differentials - ExampleMTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.11Linearization and DifferentialsCopyright © 2010 by Ron Wallace, all rights reserved.Zooming in on a Function22)(23 xxxxfx  [-3, 3]y  [-2, 5](1,2)Zooming in on a Function22)(23 xxxxfx  [0, 2]y  [-2, 5](1,2)Zooming in on a Function22)(23 xxxxfx  [0.5, 1.5]y  [-2, 5](1,2)Zooming in on a Function22)(23 xxxxfx  [0.9, 1.1]y  [-2, 5](1,2)As you zoom in on a function at a point of continuity,the graph tends approach a line.Therefore, a function can be approximatednear a point by the tangent line at that point.Linearization•Using the tangent line to a curve at a point as an approximation of a function near that point.AKA: Standard Linear ApproximationNOTE: Linearization and finding the tangent line to a function at a point … is the same thing!•If the function is differentiable at x = a, then the linearization of the function at a is …•Purpose of Linearization?It provides a quick and easy approximation of a complex function near a known point.( ) ( ) '( )( )L x f a f a x a= + -Preview of things to come …•Linearization approximates a function with a line through a point of the function (a,f(a)).•In Chapter 10, we’ll see how to …approximate a function with a quadraticapproximate a function with a cubicetc … notice the pattern?2 3''( ) '''( )( ) ( ) '( )( ) ( ) ( )2! 3!f a f aC x f a f a x a x a x a= + - + - + -2''( )( ) ( ) '( )( ) ( )2!f aQ x f a f a x a x a= + - + -( ) ( ) '( )( )L x f a f a x a= + -Preview of things to come …•Example: f(x) = sin x @ a = /42 32 2 2 2( )2 2 4 4 4 12 4C x x x xp p p� � � � � �= + - - - - -� � � � � �� � � � � �22 2 2( )2 2 4 4 4Q x x xp p� � � �= + - - -� � � �� � � �2 2( )2 2 4L x xp� �= + -� �� �( ) ( )( ) ( )( ) ( )( ) ( )24 4 224 4 224 4 224 4 2sin' cos'' sin''' cosffffp pp pp pp p= == ==- =-=- =-Preview of things to come …•Example: f(x) = sin x @ a = /42 32 2 2 2( )2 2 4 4 4 12 4C x x x xp p p� � � � � �= + - - - - -� � � � � �� � � � � �22 2 2( )2 2 4 4 4Q x x xp p� � � �= + - - -� � � �� � � �2 2( )2 2 4L x xp� �= + -� �� �( ) ( )( ) ( )( ) ( )( ) ( )24 4 224 4 224 4 224 4 2sin' cos'' sin''' cosffffp pp pp pp p= == ==- =-=- =-Preview of things to come …•Example: f(x) = sin x @ a = /42 32 2 2 2( )2 2 4 4 4 12 4C x x x xp p p� � � � � �= + - - - - -� � � � � �� � � � � �22 2 2( )2 2 4 4 4Q x x xp p� � � �= + - - -� � � �� � � �2 2( )2 2 4L x xp� �= + -� �� �( ) ( )( ) ( )( ) ( )( ) ( )24 4 224 4 224 4 224 4 2sin' cos'' sin''' cosffffp pp pp pp p= == ==- =-=- =-Preview of things to come …•Example: f(x) = sin x @ a = /42 32 2 2 2( )2 2 4 4 4 12 4C x x x xp p p� � � � � �= + - - - - -� � � � � �� � � � � �22 2 2( )2 2 4 4 4Q x x xp p� � � �= + - - -� � � �� � � �2 2( )2 2 4L x xp� �= + -� �� �( ) ( )( ) ( )( ) ( )( ) ( )24 4 224 4 224 4 224 4 2sin' cos'' sin''' cosffffp pp pp pp p= == ==- =-=- =-Preview of things to come …•Example: f(x) = sin x @ a = /42 32 2 2 2( )2 2 4 4 4 12 4C x x x xp p p� � � � � �= + - - - - -� � � � � �� � � � � �22 2 2( )2 2 4 4 4Q x x xp p� � � �= + - - -� � � �� � � �2 2( )2 2 4L x xp� �= + -� �� �( ) ( )( ) ( )( ) ( )( ) ( )24 4 224 4 224 4 224 4 2sin' cos'' sin''' cosffffp pp pp pp p= == ==- =-=- =-Along the function:Differentials( )y x dx+( )f x x+Df(x)Tangent Line:0 0 0( ) ( ) '( )( )y x f x f x x x= + -xy( ) ( )y f x x f xD = +D -xx x+D )(xyxf Along the tangent line:x dx+dx)()( xydxxydy dy0'( ) limxyf xxD �D=Ddymdx=& are used w/ the function & are used w/ the tangentx y dx dyD DSAMEDifferentials( )y x dx+( )f x x+Df(x)Tangent Line:0 0 0( ) ( ) '( )( )y x f x f x x x= + -xyxx x+D )(xyxf x dx+dxdy & are cal Differeled ntialsdx dy '( )y dy f x dxD � =dxdyxf )('Differentials•Examples …Determine dy for each of the following functions.'( )dy f x dx=3( ) 5 1f x x x= - +( ) sin5f x x=( ) lnxf x e x=dy is an estimate of the change in the function at x when there is a given change in x (called dx).Differentials - Example•The diameter of a tree was 10 inches. During the following year, the circumference increased by 2 inches. About how much did the tree’s diameter increase? About how much did the tree’s cross-sectional area increase?2 , circumferencedC C= =Use , diameterC D Dp= =2Use , area2DA Ap� �= =� ��