MTH 251 – Differential Calculus Chapter 3 – DifferentiationSecants of a CurveTangents of a CurveSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13The Derivative at a PointSlide 15The following are the same …MTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.1Tangents andthe Derivative at a PointCopyright © 2010 by Ron Wallace, all rights reserved.Secants of a Curve•SecantA line that contains two points of a function.•Example:Find the slope of the secant line of the function f(x) = x2 – 3 for the two points (1,–2) and (–3, 6).Find the equation of the secant line.Graph the function and the secant line.Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.QPf(x)Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.QPf(x)Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.QPf(x)Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.QPf(x)Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.QPf(x)Tangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.P(x0, f(x0))Q(x1, f(x1))Let h = x1 – x0Find the limit•Q  P•x1  x0•h  0QPf(x)x1x0hQPf(x)x1x0hTangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.P(x0, f(x0))Q(x1, f(x1))Let h = x1 – x0Find the limit•Q  P•x1  x0•h  0•Slope of the secant?1 01 0( ) ( )sf x f xmx x-=-0 0( ) ( )f x h f xh+ -=QPf(x)x1x0hTangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.•Slope of the secant?•Slope of the tangent?0 0( ) ( )sf x h f xmh+ -=0 00( ) ( )limthf x h f xmh�+ -=QPf(x)x1x0hTangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.•Slope of the secant?•Slope of the tangent?•Equation of the tangent?0 0( ) ( )sf x h f xmh+ -=0 00( ) ( )limthf x h f xmh�+ -=0 0( ) ( )ty f x m x x= + -QPf(x)x1x0hTangents of a Curve•The tangent line at the point P on a function f(x) is the limit of the secant line through P and Q (also on the function) as Q approaches P.•Example•Slope of the tangent?•Equation of the tangent?( )2( ) 31, 2f x xP= --0 00( ) ( )limthf x h f xmh�+ -=QPf(x)x1x0hTangents of a Curve•One more common type of problem …When is the slope of this function equal to 1.When is the slope of this function equal to 0. 2( ) 3f x x= -0 00( ) ( )limthf x h f xmh�+ -=The Derivative at a Point0 00( ) ( )limthf x h f xmh�+ -=( )0 000( ) ( )' limhf x h f xf xh�+ -=Difference QuotientGives the average rate of change of the function between the two points where x = x0 and x = x0+hThe Derivative, f ’(x0) gives the instantaneous rate of change of the function when x = x0The Derivative at a Point•ExamplesFind the derivatives of the following functions at the given values of x0.( )0 000( ) ( )' limhf x h f xf xh�+ -=01( ) , 2f x xx= =0( ) , 9f x x x= =The following are the same …•The Derivative of f(x) at x = x0•The Limit of the Difference Quotient of f(x) at x = x0•The Slope of the Tangent Line to y = f(x) at x = x0•The Slope of y = f(x) at x = x0•The Rate of Change of f(x) wrt x at x = x00 00( ) ( )limhf x h f xh�+