微積分常用公式

1、16.d T/Wx|_ g(x)微積分常用公式及常用述語中英文對照一、基本微分法則L頭Y，疔爲(wèi)常數(shù)axS=曲、応R dx3. ? 1/00 士呂二?/（或士?匱axaxax紅? fMsM 二f（力或?qū)? fg （對dx5- 土1/（斗）嵐（兀）城盂）卜八小呂（龍護(hù)+/（初才（兀）血（尢）十/（尢）呂為S）二、三角函數(shù)之微分1.sin & - cos 92. cos 9= sni 9 d826.蘭 csc-i 耳=.-zL_ 必k|g-i5. sec 9= tan Osec 0 dB6. -cEce = -cotecsce4.二、心dx1 + x2四T旨數(shù)函數(shù)與對數(shù)函數(shù)之微分 d v三、反三角函

2、數(shù)之微分X 02. F(x + l)=zr(x)3.廠(科 +1) = 1A、Beta函數(shù)1. |3(戸衛(wèi))=J：疋J(l-x)E必 J pOrq 1)52+1(2 4-1)!九、施轉(zhuǎn)體之體積L薄片法：f，其中力爲(wèi)立龍?jiān)诓分械拿恳回Ｌ幋怪膘懂嬢S之截面積2.囿盤法：7 =兀嚴(yán)（勸僉3.墊圈法：滄咱嚴(yán)弋乞加4圓柱殼法:卩=2耐g十、孤長L =打1 + 八）咯 十一、泰勒級數(shù)L金“打3+響（+即）+2 + + 223.山亍亍+（莎十二、長見的馬克勞林輾數(shù)6中示=生in4. cosh x = -二 1 + + +、* + -+ -、葢已 R22J 4!(加)|5* = 1 -+ X2 - Z3 + J

3、L+ - Z5 + 3 -1 11+X氐叫1+對=訐百虧+1M紳2令+務(wù)冷+一JCXjjr8.唄一“ +三+ 了 + 丁 +一十三質(zhì)量與質(zhì)心坐標(biāo)L若障片爲(wèi)均勻密度1該薄片的面積爲(wèi)丄，則其質(zhì)量爲(wèi)p4。2.若薄片爲(wèi)非均勻密度，區(qū)城尺的面積密度函數(shù)爲(wèi)災(zāi)J）:則其總質(zhì)量爲(wèi) JJp（扎刃 4。3.區(qū)域氏爲(wèi)非均勻密度，其面積密度函數(shù)爲(wèi)則其質(zhì)心坐標(biāo)爲(wèi)（乳刃 其JjpG丿)曲R _p(xty)dAA A _JJ p,y)dA710,十四、積分表 （一）張対多項(xiàng)式型1. J滋二戀M(fèi)Jr一出 v = Mn 兀 + C5. J妙（x）必二町/必6J/（M 土鼠陸訂/（對必土 Jg陸（二八三角函數(shù)型sin xdx-

4、- cos 開 + Ucos Mr 二 sin x + C4. J cot = n |stn x + C5. J sec xdx - n |sec x + tan x| 4- (76. Jcscxdx = h|cscx - cot x 4-C7. J sec xtan xdx = sec x + C8. esc x cot xdx = - esc x + C7,sin 2x cx +asin 曲疋)+ Ctn +n+ _-_+u耳(擷-可+加)2 護(hù)(旳一 1)+肚)昭一1-4.- -必=InJ x(a + 處) a5* -片dx - -_ AinJ x(a +ix) a(a +bx) a1

5、, & -（六） 為+如型1. xjn + 加必=(j - #r)JS +&兀尸 + 匚 J5i? 15o+ C133, |必=2jm +&忑 + a-十 j-ax2a J +bx1, J J/ /djr =(兀 J?2rjrj2 2 InIn x x + +/ ) + C3.4.5.6.7.“必二 +/ - a0n + x/xJ*-2-i H 門-dx -寸護(hù)-a2 - a sec 1 + CJTa-f 1 一任=In z 卡 JF 土/ 4- C 士1-1dx - (h a + J 衛(wèi) +a+ tjIn |x|) + Ct_ -J-_11_2* J x2 Jdr +抓必=x2 J(a +

6、加-2(ar J x Ja + bxJ dx+ bxj， dx = = On l(a +Zx - | - In+加 + 4 加 Va 111i 1 t - & +加 -b5., . dx-z w + 肚（七）、JF 土/ 型Jiadx + (J? ti2) + C2 I24J/ J* 土皿臥-玄空 蘭,)土/ - In z + 土 / ) + C山+ Ua1.2.3.4r 聽了 - F必=+ sin-1 4- CJ22 a心匸7必=空亡旦血匚F + duiV + C188 a-dx - /a2 -m(bi 訛十-丿 xr腫一 F才匚疋 “ x- -ax = - sin 一 十 C71T穴-ln

7、|xj) + C14155.(In + y/ct2 j3 In 贋J) + C8.血嚴(yán)=_圧& +(AD、/士/型1,1_1 z X. 三-y t/x tan () + jx +a a a1 &蓋 _x+2-3 r(a2 X2)K_ Z?(科一 l)(f?H)m2i?(1) J (a2 x2)1.2.3.x- a1 =J/X = sm_1- + C F復(fù)1、心英-a2 - x2 a微積分常用述語中英文對照AChin eseabsolute maximum絕對最大值absolute mini mum絕對最小值absolutely con verge nt絕對收斂absolute value絕對值

8、accelerati on加速度alter nat ing series交錯(cuò)級數(shù)an gle角an tiderivatives反導(dǎo)數(shù)approach趨近arc len gth弧長argume nt輻角16average value平均數(shù)BChin esebou nded above有上界的bou nded below有下界的bou nded seque nee有界數(shù)列CChin eseealeulus微積分cardioid心臟線cen ter of mass質(zhì)心Chai n Rule鏈?zhǔn)椒▌tclosed in terval閉區(qū)間coefficie nt係數(shù)Complex n umbers複數(shù)C

9、on clusi on( delete)結(jié)論(delete)con diti onal in equality條件不等式coord in ate座標(biāo)coord in ate axis坐標(biāo)軸curvature曲率copla nar共平面conic secti on二次曲線Complete ness Axiom完備性公理con tradictory lin ear equati ons矛盾方程組cycloid擺線con diti on ally con verge nt條件收斂critical n umber臨界值conv erge收斂cyli nder圓柱體，柱體cross-secti on橫切

10、面curve sketchi ng曲線描繪curve曲線con cave dow nward凹向下con cave upward凹向上17composite fun cti on合成函數(shù)DChin esedomai n定義域denomin ator分母derivative導(dǎo)數(shù)depe ndent lin ear equati ons相依方程式discrim inant判別式differe nee set差集divide nd被除數(shù)divisor除數(shù)diverge nt發(fā)散differe ntiati on微分disc on ti nuity不連續(xù)decay衰變decimal小數(shù)decreas

11、ing遞減dime nsional維數(shù)dot product點(diǎn)積directi on an gle方向角directi on cos ine方向餘弦determ inant行列式directi on n umber方向數(shù)de nsity密度double in tegrals雙重積分EChin eseexponent指數(shù)exp onen tial law指數(shù)率exp onen tial fun cti on指數(shù)函數(shù)extreme values of fun cti ons函數(shù)的極值expressi on函數(shù)表示法eve n fun cti on偶函數(shù)empty set空集合equati on方程

12、式18ellipse橢圓ellipsoid橢球FChin eseFib on acci seque nee費(fèi)伯納希數(shù)列four-leaved rose四葉玫瑰fun cti on函數(shù)Ggraphical soluti on圖解法Gaussia n fun cti on高斯函數(shù)geometric series幾何級數(shù)global maximum全局最大值global mi ni mum全局最小值gradie nt斜度，變化率HChin esehalf-life半衰期harm onic fun cti on調(diào)和函數(shù)harm onic series調(diào)和級數(shù)helix螺旋線horiz on tal a

13、symptote水平漸進(jìn)線hypote nuse斜邊hypothesis假設(shè)hyperbola雙曲線hyperbolic fun cti on雙曲函數(shù)IChin eseimplicit differe ntiati on隱函數(shù)微分implicit fun cti on隱函數(shù)improper in tegral瑕積分in creas ing遞增indefinite integral不定積分19in determ in ate form不定型infinite series無窮級數(shù)in flecti on point反曲點(diǎn)in itial point起點(diǎn)inner product內(nèi)積In sta n

14、tan eous velocity瞬時(shí)速度in tegrable可積分in tegrati on積分in tegra nd被積函數(shù)in tegrati on by parts分部積分in tercept截距in terval of con verge nee收斂區(qū)間in verse fun cti on反函數(shù)In verse trigo no metric fun cti on反三角函數(shù)infinity無限大irrati onal n umber無理數(shù)in fere nee推理in terval區(qū)間in tegral exp onent整數(shù)指數(shù)JChin eseKChin eseLChin e

15、seL Hospital s Rule羅必達(dá)法則Laplace s equation拉普拉斯方程式Law of Cosines餘弦定理least upper bou nd最小上界left-hand limit左極限lin ear in equality in two unknown variable二元一次不等式20limit極限Limit law極限法則lin ear approximati on線性逼近lin ear equati on線性方程式lin earizati on線性化local maximum局部最大值local mi ni mum局部最小值logarithmic Fun c

16、ti ons對數(shù)函數(shù)lower limit下限MChin eseMaclaurin series馬克勞倫級數(shù)maximum極大值matrix矩陣Mean Value Theorem均值定理midpo int rule中點(diǎn)法則minimum極小值mome nt彎矩mome nt of in ertia轉(zhuǎn)動(dòng)慣量,慣性矩monotonic單調(diào)multiple倍數(shù)NChin eseNewto ns Method牛頓法no rmal li ne法線no rmal vector法向量n umerator分子OChin eseodd fun cti on奇函數(shù)On e-to-one fun cti on一對

17、一函數(shù)ope n in terval開區(qū)間optimizati on problem最佳化問題orthogo nal正交21origin原點(diǎn)PChin eseparabola拋物線parameter參數(shù)parametric curve參數(shù)曲線parametric equati on參數(shù)方程式partial derivative偏導(dǎo)數(shù)partial fracti on部份分式period週期periodic fun cti on週期函數(shù)perpe ndicular垂直polar axis極軸polar coord in ate極座標(biāo)polar coord in ate system極座標(biāo)系統(tǒng)po

18、lar equati on極方程式polar mome nt of in ertia極性轉(zhuǎn)動(dòng)慣量property極點(diǎn)polyno mial多項(xiàng)式polyno mial fun cti on多項(xiàng)函數(shù)power series幕級數(shù)projecti on投影p-seriesp級數(shù)QChin esequadric surface二次曲面quotie nt商RChin eseradius of con verge nee收斂半徑radia n弧度rati onal fun cti on有理函數(shù)rati onal fun cti on有理函數(shù)real n umber實(shí)數(shù)22relative error相對誤差relative maximum相對最大值range值域rema in der餘數(shù)Riema nn sun黎曼和right-ha nd limit右極限r(nóng)oot fun cti on根函數(shù)SChin esesaddle point鞍點(diǎn)scalar純量scalar product純量積sec ond mome nt同 moment of inertiasequenee數(shù)列series級數(shù)Simps on s Rule辛普森法sla nt asymptote斜漸進(jìn)線slope斜率solid of revoluti on旋轉(zhuǎn)體soluti on set解集合