AP微积分-APCalculus公式大全-217

1、AP Calculus BCI. LimitL ImportNnt limitslim.TT()lim.TTOCsinxx=1、limx-xLHopihds Rulesin ax lim5 bx洛必达法则（#/0,hn m n/0 8 / 0X /O0)oOGiven two differentiable functions f (x) and g(x),lim f(x) = lim g(x) = 0 .thenX-X7A-W7limZ = limZV25 g(x) 5 gG)hmf(x) = limg(x) = 00 .thenX-XT.YTdfWII. Derivative1. Basic

2、 Formulas(C)r = 0 (C is a constant)(xn)f = nxn(exy = ex. (axY = ax Inn (a O.a * 1)(In# = -: (log& xY = i (a 0,“ 工 1) xxma(sill x= cosx(cosx)*= -sinx(tan x/= sec2 x(cot)r= -esc2 x(secx)= sec x tan x(6) (arcsin x)1 =,】(arctanx)*= !1+f(cscx)= -cscxcotx(arccos x)* = -(flrcotx) = -1 +对(tnrsecx)|a-|V?T(“r

3、cscQ= |.v|V7T2. Rules(1) Ify(x), g(x) are differentiaba. g)g(x)y = f(兀)g3 :b. (f(x)g(x)y = fXx)gx) + f(X)gXx)t especially, (Cf(x)Y = Cfx) (Cisa constant)：c. (竺mm)小) g(X)gl(如0), es(册晋dxdy(5) Parametric fonction dydx(3) Implicit Differentiation(4) Inverse fimction jv 1:= dx dxdx dx7W wh(“i thagg3. App

4、lications of DQriztivQ(1) Mean Value TlieoremSign of f1+MonotonicityIncreaseDecrease(3) Properties of curvesCritical point: f x) = 0 or DNESign of 厂+ConcavityConcaveConcaveupdownPoint of inflection: concavity changes.(2) Cham Rule y =些 dx(6) Polar Action冬=广血外“翻 dx广(&)cos&-厂 sin&(7) Vector (r(7j)* =

5、(/VXg V)b-aGiven f(x) is differentiable, f (c) = 11_, where c e (a, b).(2) Tangent and normalSlope of tangent Slope of nonnal = -1Linearization: /(x) a L(x) = /() + f ci)(x a).Second Dem atrve Test f n(a) 0 , local mui f a) J-x2r 1dx = arctan x + CJ 1 + x2I ( dx = arcsecx + Cxx2 U-substitiition (常见u

6、的选择分母,Steps: 1. Let u=u(x), find dll = u x)dx,f“ dx = arcsin + Cf -dx = arctan + C aaf= arcsec + CJa a底数,指数，角度等)7(2心)“G)厶= Integration by partsudv = MV,(or dx = uv-vudxTabular Integration (story about liu sinx, and ex )Mean Value theorem: (average alue of f (x) over1 fba. b) / (c) = f(x)dx .where c

7、e (, b).b-aJa 3ImptopQT IntQgml反常积分(两种形式，无穷积分，瑕积分) Integral on infinite intervalf /(x)JA = lim f(x)dx,ax J af f(x)dx = lim f(x)dxyJ *ctoo J af f(x)dx = f f(x)clx+ f f(x)clx= lim f(x)dx + hm f f(x)dx.J00 *JOC *JcdT-OoJd *X Jc Integrand with mfimte discontinuitiesf(x)dx = lim f(x)dx.f(x)dx = lim f(x)d

8、x,b(tcfib fMdx = f(x)dx + J fWdx = Jiin f(x)dx + im f(x)dx.4. RulesRiemann Sum (LRANL RRAM, MRANL Trapezoidal Rule)Q (x) + /g(x)dx = q /(jv)必+ /J； g(x)dx, fWdx = f(x)dx + f f(x)dxj7(x)J.r = 0J： f (x)dx =f (x)dx,f(xdx = 2 fx)dx (f (x)is even) -dJof(x)dx = 0(/(x)is odd)5. TheoremFundamental Tlieorem o

9、f Calculus 微积分基本宦理 厂 = 3： =/()一/， or/(b) = /(d) + f 厂気 fg=fg,6. Application of IntagmlGeneral form：牛 f ： f(t)dt = /(%) a (x) - f(0(x) 0G)八Vblume： witli known cross sectionr(tg(x)dx、Horizontal Slicef(y)-g(y)_ upper functionlower function _ Right (uiKtionLcl !unclion_bArea： Vertical Slice |Ax)dx , A(x

10、) is die area of the cross section.Revolution： 7r R _ 厂Ja LOulcr radius Inner radiusShell Method： 2/rJ x)yA(旋转轴为 y-axis) or 2/rJ xydy (旋转轴为 x-axis)LengthIV. DiffQTnti:d Equation1.SeparNtion VariablQ =)=dx N(y)2.Logistic Equationw二 Jim P(t) = K ,where K is the em-iromneiital + e r 00capacity.3. Slope

11、 Fidd4. EulWs Method：1. Definition2. Tast for ComagQncQAssume tliat the following limit exists: p = limIl-XXZX =4+2 + + %+, Ratio Test00V an ConvergesO lim Sn = S 厶千“TOCXa) If pel, then 为an converges absolutely./i-i00b) If p 1, then 丫 cin diverges.;r-ic) If p = 1, then the test is inconclusive (the

12、series may converge or diverge). Integral Test Let Cln = f (fl) ,where f (x) is positive. decFeasiug and coHthmous for X 1.Both f(x)dx and，f converges or both diverges (同时收敛或同时发散，但是不相等). “1 Comparison TestAssume tliat there exists M 0 such that 0 an M.If f”】converges, then If 工/!-lalso converges.XXI

13、f 工勺】 diverges, then 工仇 alson-in-!diverges.Lmiit Comparison Test Let%and hnbe positive sequences. Assumelim H = LTO bIfXXL 0 then 工 converges if and only if 工 bn converges, n-lM-lb)If8X厶=s, andan converges, then 工 bnn-ln-lconverges.c)IfXXL = 0, and工bn converges, thenancom*erges.n-l;r-l3. Four Series

14、The geometric series a,J comerges if I r l 1 and diverges otherwise, /r-l U.二 1Harmonic Series Diverges.x (_1)Alternating Series terms are alternately positive and negative, e.g.工 Lcbmz Test fbr Altematmg Series Assume that ;/ is a positive sequence that is decreasmg andconverges to 0: q % 0, lim an

15、 = 0.-moc=0 +a (x-c) + 2 (x_c)- +q (x_c) +x Power Series 工an (x-c)n-0(i) Radius of ConvergencexEvery power series F(x)=a” (x_c) has a radius of convergence R , which is eitlier a nonnegath-e number ( /? 0 )or infiiiity( /? = oo).If R is finite, F(x) conveTges absolutely when |x c| R IfR = 8, F(x) co

16、m-erges absolutely for all X .DivergesConverges absolutelyDivergesc-Rcc-RPossible convergence at the endpoints(ii) Tenn-bv-term Differentiation and IntegrationAssume that F(x)=工(X c) has radius of convergence 7? 0 . Then is differentiable on /r-b且”=i a i= O)SCO) 负数没有偶次方根：0的任何次方根都是0,记作”5 = 0。当是奇数时，0

17、= 0,当是偶数时,2. 分数指数幕正数的分数指数幫的意义，规定：m atl = (a Ojnji e Nn 1) ain1an 0的正分数指数慕等于0, 0的负分数指数幕没有意义3.实数指数幕的运算性质（aO/seR）1 (a 0,m.n e Nn 1)（1） ar ar =r+v： （2） S） =a八:（3）（） =ara5（二）指数函数及其性质1、指数函数的槪念：一般地，函数y = 0”（40,且。工1）叫做指数函数，英中x是自变量，函数的 泄义域为R注意：指数函数的底数的取值范用.底数不能是负数、零和12、指数函数的图象和性质al0a0值域y0在R上单调递增在R上单调递减非奇非偶函数

18、非奇非偶函数函数图象都过泄点（0, 1）函数图象都过建点（0, 1）二、对数函数（）对数1. 对数的概念：一般地，如果a”=N（aO,dHl）,那么数x叫做以a为底的对数，记作: x = oaN （一底数，N真数，logN 对数式）说明：注意底数的限制d0,且dHl两个重要对数：常用对数：以10为底的对数lg/V:（2）自然对数：以无理数 = 2.71828为底的对数InTV.指数式与对数式的互化幕值 真数a = NU loga N = b底数log, = 指数对数（-）对数的运算性质如果a0.且dHl, M0, N 沁 那么:loga（M N） = log M + logn N :log為=

19、 logM - log“N :log幺 M = n log M （fi e R）换底公式 log b = （a0,且 dHl； c0,且 CH1： /?0 ）.log,利用换底公式推导下而的结论 log bn = logj?：m对数恒等式ah =b（二）对数函数1、概念：函数y = logflx（6/0,且。工1）叫做对数函数，其中x是自变量，左义域是（0, +8）. 对数函数对底数的限制：（a0,且dHl）.2、对数函数的性质：al0al0定义域x0值域为R值域为R在R上递增在R上递减函数图象都过 定点（b 0）函数图象都过定点（1, 0）（三）幕函数1、幕函数泄义：一般地，形如y = （a

20、 e /?）的函数称为幕函数，其中a为常数.2、泵函数性质归纳.（1）所有的幕函数在（0, +8）都有定义并且图象都过点（1, 1）：（2）G0时，幕函数的图象通过原点，并且在区间10,+00）上是增函数.特别地，当Q1时，幕函数 的图象下凸：当0vavl时，幕函数的图象上凸；（3）GV0时，幕函数的图彖在区间（O.+oo）上是减函数.在第一象限内，当X从右边趋向原点时，图象 在y轴右方无限地逼近y轴正半轴，当x趋于+s时，图象在x轴上方无限地逼近x轴正半轴.三角函数基本知识点小结一.六种三角函数yXsin = cos& = rrtan = 2丄*yrrsec& = esc。=xy二.同角三角函数基本关系式 i.倒数关系2.商的关系sin a csca = 1:cosa seca = 1secacosa csca；cota =3.平方关系tana-cota = l：sin atan a =cosa cscasin a sec asin2 a + cos2 a = 1 ； 1 + tan2 a = sec2 a :1 + cot2 a = esc2 a三.两角和与差，二倍角公式 sin(a 0) = sin a cos p cos a sin p,c、 tana 土 tan0tan a(a 0)=1 + tan