$2.MathematicalPreliminaries1 The Derivative of Function Differential CalculusAny Physical law can be represented by the function of variables1For example :$= s(t) = So + vot + at?2slopeSFree fall:s= s(t)=,gr’ parabolaFigure we know velocity(v-gt)Problem:How to get velocity from s(t)1O t.The existenceness of siopIn geometry it is slope of the function. Tshould be proved in Math courseDefinition:Ass(t) - s(to) _ ds(t)limv(to) = lim Limit;dtNr→0 △tt->1ot-toDerivative;3ds(t) differential,infinitesmal
§2.Mathematical Preliminaries Any Physical law can be represented by the function of variables. For example : Free fall: parabola Figure we know velocity(v=gt) Problem:How to get velocity from s(t) In geometry it is slope of the function. Definition: Limit; Derivative; ds(t) differential,infinitesmal 2 0 0 2 1 s = s(t) = s + v t + at 2 2 1 s = s(t) = gt dt ds t t t s t s t t s v t t t t ( ) ( ) ( ) ( ) lim lim 0 0 0 0 0 = − − = = → → s 0 t0 t slope The existenceness of siop should be proved in Math course. 1.The Derivative of Function Differential Calculus
(3x+2)(x-1) =53x2 - x - 2lim f(x) - limLimit: y= f(x) =x-1X-1x-x-1Forn ulas for simple functions① y=f(x)=CC-Cf(x+△x)y =f(x)= lim=0lim.ArArAxr-0Ax->0@y= f(x)=xAx(x+△x)- xy =f(x)= limlimArAr-→>0Ar-0Ax2③y=x(x + Ax)2 - x2 limlim (2x + △x) = 2xy=AxAr->0△r-→0
Limit: 1 3 2 ( ) 2 − − − = = x x x y f x 5 1 (3 2)( 1) lim ( ) lim 1 1 = − + − = → → x x x f x x x Formulas for simple functions y = f (x) = C lim 1 ( ) ( ) lim 0 0 ' ' = = + − = = → → x x x x x x y f x x x x x x x x x x y x x lim (2 ) 2 ( ) lim 0 2 2 0 ' = + = + − = → → y = f (x) = x ( ) lim lim 0 0 ( ) 0 ' ' = − = = = = → + → x c c y f x x x f x x x 2 y = x