2014-AP微積分CALCULUS知識點(diǎn)總結(jié)

1、AP CALCULUS Derivative 1A DERIVATIVE FUNCTION1.The derivative function or simply the derivative is defined as )(xf yxxfxxfxyxx)()(limlim002.Find the derivative functiona) Find ,yb) Find the average rate of change ,xyc) Find the limit .xyx0lim3.Geometric significanceConsider a general function y=f(x)

2、, 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, is .axafxfax)()(lim)(af Thus, we can know the derivative at x=a is the slope of the tangent at x=a.4.Rules)(xf)(xf C(a constant)0nx1nnxxsinxcosxc

3、osxsinxtanxx22cos1secxarcsin2-11xAP CALCULUS Derivative 2xlnx1xalogexalog1xexexaaaxln)()(xvxu)()(xvxu)()(xvxu)()()()(xvxuxvxu)()(xvxu 2vvuvu)0( v5. The chain ruleIf where then .)(ufy )(xuu dxdududydxdy )()(xgexf)()()(xgexfxg )(ln)(xgxf)()()(xgxgxf, )(ln)()(ln)()()()(xuxvxuxveexuxfxv)()()()(ln)()()(l

4、n)(xuxuxvxuxvexfxuxv6. Inverse function, Parametric function and Implicit functionInverse function:, , dydxdxdy1 )(1)(1xfxfi.e., , xyarcsinyxsinAP CALCULUS Derivative 3Parametric function:, dtdxdtdydxdy i.e., , )(ty)(tx)(1xt)(1xy)()(ttdtdxdtdydxdtdtdydxdyImplicit function: , .0)(,(xyxF0)(,(xfxF, , t

5、0-222ayxtaytaxsincos2 , 0ttatadxdyxycotsincos)(7. High derivativexxfxxfdxydxfx )()(lim)(022tatatdtdxdtyddxydxyxyx32sin1sincsc )()( xxfxxfxfnnxn)()(lim)()1()1(0)(y=sinx , )2sin(cosxxy)22sin()2cos( xxy)2sin()(nxynB APPLICATIONS OF DIFFERENTIAL CALCULUS1. Monotonicitya) If S is an interval of real numb

6、ers and f(x) is defined for all x in S, AP CALCULUS Derivative 4then： f(x) is increasing on S for all x in S, and0)( xf f(x) is decreasing on S for all x in S.0)( xfb) Find the monotone interval Find domain of the function, Find , and x which make ,)(xf 0)( xf Draw sign diagram, find the monotone in

7、terval.2. Maxima/Minima, Horizontal inflection, Stationary pointC INTEGRAL1. The idea of definite integralWe define the unique number between all lower and upper sums as AP CALCULUS Derivative 5 and call it “the definite integral of from a to b”,badxxf)()(xfi.e., where .niinibaixxfdxxfxxf110)()()(na

8、bxWe note that as , andn10)()(nibaidxxfxxf baniidxxfxxf)()(1We write .baniindxxfxxf)()(lim1If for all x on a,b then 0)(xf is the shaded area.badxxf)(2. Properties of definite integralsbabadxxfdxxf)()(, c is any constantbabadxxfcdxxcf)()(cabacbdxxfdxxfdxxf)()()(AP CALCULUS Derivative 6bababadxxgdxxfd

9、xxgxf)()()()(, where )()()()(aFbFxFdxxfbabadxxfxF)()(（f(x) odd）,(f(x)even)aadxxf0)(aaadxxfdxxf0)(2)(If on then 0)(xfbxabadxxf0)(If on then )()(xgxfbxababadxxgdxxf)()(The average value of a function on an interval a,bbaavedxxfabf)(13. The infinite integralIf , then )()(xfxFCxFdxxf)()(Formulas:, Cxndxxnn111Caadxaxxln1,CxinxdxcossCxxdxsincos,CxxdxcoslntanCxxdxs