学习必备欢迎下载A DERIV ATIVE FUNCTION 1.The derivative function or simply the derivative is defined as )( xf＝ y ＝xxfxxfxyxx)()(limlim002.Find the derivative function a) Find y, b) Find the average rate of change xy , c) Find the limit xyx0lim. 3.Geometric significance Consider a general function y=f(x), a fixed point A(a,f(a)) and a variable point B(x,f(x)). The slope of chord AB=axafxf)()(. Now as BA, xa and the slope of chord ABslope of tangent at A. So, axafxfax)()(limis )(af. Thus, we can know the derivative at x=a is the slope of the tangent at x=a. 4.Rules )( xf)( xfC(a constant) 0 nx1nnxxsinxcosxcosxsinxtanxx22cos1secxarcsin2-11x学习必备欢迎下载xlnx1xalogexalog1xexexaaax ln)()(xvxu)()(xvxu)()(xvxu)()()()(xvxuxvxu)()(xvxu2vvuvu)0(v5. The chain rule If )(ufywhere )(xuuthen dxdududydxdy. )()(xgexf)()()(xgexfxg)(ln)(xgxf)()()(xgxgxf)(ln)()(ln)()()()(xuxvxuxveexuxfxv, ])()()()(ln)([)()(ln)(xuxuxvxuxvexfxuxv6. Inverse function, Parametric function and Implicit function Inverse function:dydxdxdy1, ])([1)(1 xfxf, i.e., xyarcsin, yxsin学习必备欢迎下载Parametric function:dtdxdtdydxdy, i.e., )(ty,)(tx→)(1 xt, )]([1 xy)()(ttdtdxdtdydxdtdtdydxdyImplicit function: 0))(,(xyxF, 0))(,(xfxF. 0-222ayx, taytaxsincos, t]2,0[ttatadxdyxycotsincos)(7.High derivative xxfxxfdxydxfx)()(lim)(022tatatdtdxdtyddxydxyxyx32sin1sincsc])([)(xxfxxfxfnnxn)()(lim)()1()1(0)(y=sinx )2sin(cosxxy, )22sin()2cos(xxy)2sin()(nxynB APPLICATIONS OF DIFFERENTIAL CALCULUS 1. Monotonicity a) If S is an interval of real numbers and f(x) is defined for all x in S, then：学习必备欢迎下载f(x) is increasing on S 0)(xffor all x in S, and f(x) is decreasing on S 0)(xffor all x in S. b) Find the monotone interval Find domain of the function, Find )(xf, and x which make 0)(xf, Draw sign diagram, find the monotone interval. 2. Maxima/Minima, Horizontal inflection, Stationary point C INTEGRAL 1. The idea of definite integral We define the unique number between all lower and upper sums as badxxf)(and call it “the definite integral of )(xffrom a to b”, 学习必备欢迎下载i.e., niinibaixxfdxxfxxf110)()()(where nabx. We note that as n, 10)()(nibaidxxfxxfand baniidxxfxxf)()(1We write baniindxxfxxf)()(lim1. If 0)(xffor all x on [a,b] then badxxf)(is the shaded area. 2. Properties of definite integrals babadxxfdxxf)()]([babadxxfcdxxcf)()(, c is any constant cabacbdxxfdxxfdxxf)()()(bababadxxgdxxfdxxgxf)()()]()([学习必备欢迎下载)()()()(aFbFxFdxxfbaba, where dxxfxF)()(aadxxf0)(（f(x) odd）,aaadxxfdxxf0)(2)((f(x)even) If 0)(xfon bxathen badxxf0)(If )()(xgxfon bxathen babadxxgdxxf)()(The average value of a function on an interval [a,b] baavedxxfabf)(13. The infinite integral If )()(xfxF, then CxFdxxf)()(Formulas:Cxndxxnn111, Caadxaxxln1Cxinxdxcoss,Cxxdxsincos,Cxxdxcoslntan,CxxdxsinlncotCxxdxarcsin12(12x), Cxxdxarctan12U Substitution dxxgxgf)())((substitution u=g(x) duuf)(Integration by Parts vduuvudv学习必备欢迎下载
