《微積分英文版》ppt課件

1、CHAPTER 3THE DERIVATIVE微積分學(xué)的開創(chuàng)人: 德國(guó)數(shù)學(xué)家 Leibniz 微分學(xué)導(dǎo)數(shù)導(dǎo)數(shù)思想最早由法國(guó)數(shù)學(xué)家 Ferma 在研討極值問(wèn)題中提出.英國(guó)數(shù)學(xué)家 Newton2.1nTwo Problems with One ThemeTangent Lines & Secant LinesnThe slope of a secant line between 2 points on a curve is the change in y-values divided by the change in x-values.nSince a tangent line touch

2、es only one point on the curve, how do we find the slope of the line? We consider the slope of 2 points that are INFINITELY close together at the point of tangencythus a limit!Average Velocity & Instantaneous VelocitynSimilar to slope of a secant line, to find average velocity, we find the chang

3、e in distance divided by the change in time between 2 points on a time interval.nTo find instantaneous velocity, we find the difference in distance and time between two points in time that are INIFINITELY close togetheragain, a limit!Tangent Line Slope at x = c & Instantaneous Velocity at t = c

4、are defined the SAMEhcfhcfvmh)()(lim0tan一、一、 引例引例1. 變速直線運(yùn)動(dòng)的速度變速直線運(yùn)動(dòng)的速度設(shè)描畫質(zhì)點(diǎn)運(yùn)動(dòng)位置的函數(shù)為)(tfs 0t那么 到 的平均速度為0tt v)()(0tftf0tt 而在 時(shí)辰的瞬時(shí)速度為0t lim0ttv)()(0tftf0tt 221tgs so)(0tf)(tft自在落體運(yùn)動(dòng)機(jī)動(dòng)機(jī)動(dòng) 目錄目錄 上頁(yè)上頁(yè) 下頁(yè)下頁(yè) 前往前往 終了終了 A falling bodys velocity is defined. Find the instantaneous velocity at t = 1 seconds.sec)/

5、963(32)1632(lim1632lim16163216lim16)2(16lim16)(16lim16020222022202202ftvtthhthhhthhthththththththttvhhhhh xyo)(xfy C2. 曲線的切線斜率曲線的切線斜率曲線)(:xfyCNT0 xM在 M 點(diǎn)處的切線x割線 M N 的極限位置 M T(當(dāng) 時(shí))割線 M N 的斜率tan)()(0 xfxf0 xx切線 MT 的斜率tanktanlim lim0 xxk)()(0 xfxf0 xx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 兩個(gè)問(wèn)題的共性:so0t)(0tf)(tft瞬時(shí)速度 lim0ttv

6、)()(0tftf0tt 切線斜率xyo)(xfy CNT0 xMx lim0 xxk)()(0 xfxf0 xx所求量為函數(shù)增量與自變量增量之比的極限 .類似問(wèn)題還有:加速度角速度線密度電流強(qiáng)度是速度增量與時(shí)間增量之比的極限是轉(zhuǎn)角增量與時(shí)間增量之比的極限是質(zhì)量增量與長(zhǎng)度增量之比的極限是電量增量與時(shí)間增量之比的極限變化率問(wèn)題機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 Rest of Change:3.2 The DerivativenThe derivative of f(x) is designated as f(x) or f or y.hxfhxfxfyh)()(lim)( 03.2 The D

7、erivativehcfhcfcxcfxfcfhcx)()(lim)()(lim)( 0思索與練習(xí)思索與練習(xí)1. 函數(shù)函數(shù) 在某點(diǎn)在某點(diǎn) 處的導(dǎo)數(shù)處的導(dǎo)數(shù))(xf0 x)(0 xf )(xf 區(qū)別:)(xf 是函數(shù) ,)(0 xf 是數(shù)值;聯(lián)絡(luò):0)(xxxf)(0 xf 留意留意:有什么區(qū)別與聯(lián)絡(luò) ? )()(00 xfxf？與導(dǎo)函數(shù)機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 二、導(dǎo)數(shù)的定義二、導(dǎo)數(shù)的定義定義定義1 . 設(shè)函數(shù)設(shè)函數(shù))(xfy 在點(diǎn)0 x0limxx00)()(xxxfxfxyx0lim)()(0 xfxfy0 xxx存在,)(xf并稱此極限為)(xfy 記作:;0 xxy; )(0

8、 xf ;dd0 xxxy0d)(dxxxxf即0 xxy)(0 xf xyx0limxxfxxfx)()(lim000hxfhxfh)()(lim000那么稱函數(shù)假設(shè)的某鄰域內(nèi)有定義 , 在點(diǎn)0 x處可導(dǎo), 在點(diǎn)0 x的導(dǎo)數(shù). 機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 運(yùn)動(dòng)質(zhì)點(diǎn)的位置函數(shù))(tfs so0t)(0tf)(tft在 時(shí)辰的瞬時(shí)速度0t lim0ttv)()(0tftf0tt 曲線)(:xfyC在 M 點(diǎn)處的切線斜率xyo)(xfy CNT0 xMx lim0 xxk)()(0 xfxf0 xx )(0tf )(0 xf 機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 0limxx00)()(xx

9、xfxfxyx0lim)()(0 xfxfy0 xxx假設(shè)上述極限不存在 ,在點(diǎn) 不可導(dǎo). 0 x假設(shè),lim0 xyx也稱)(xf在0 x假設(shè)函數(shù)在開區(qū)間 I 內(nèi)每點(diǎn)都可導(dǎo),此時(shí)導(dǎo)數(shù)值構(gòu)成的新函數(shù)稱為導(dǎo)函數(shù).記作:;y; )(xf ;ddxy.d)(dxxf就說(shuō)函數(shù)就稱函數(shù)在 I 內(nèi)可導(dǎo). 的導(dǎo)數(shù)為無(wú)窮大 .機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 3.6 Leibniz NotationDifferentiability implies continuity.nIf the graph of a function has a tangent at point c, then there is n

10、o “jump on the graph at that point, thus is continuous there.處可導(dǎo)在點(diǎn)xxf)(函數(shù)的可導(dǎo)性與延續(xù)性的關(guān)系函數(shù)的可導(dǎo)性與延續(xù)性的關(guān)系定理定理.處連續(xù)在點(diǎn)xxf)(證證: 設(shè))(xfy 在點(diǎn) x 處可導(dǎo),)(lim0 xfxyx存在 , 因此必有,)(xfxy其中0lim0 x故xxxfy)(0 x0所以函數(shù))(xfy 在點(diǎn) x 延續(xù) .留意留意: 函數(shù)在點(diǎn)函數(shù)在點(diǎn) x 延續(xù)未必可導(dǎo)延續(xù)未必可導(dǎo).反例反例:xy xyoxy 在 x = 0 處延續(xù) , 但不可導(dǎo).即機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 2. 設(shè)設(shè))(0 xf 存在 , 那

11、么._)()(lim000hxfhxfh3. 知知,)0(,0)0(0kff那么._)(lim0 xxfx)(0 xf 0k4. 假設(shè)假設(shè)),(x時(shí), 恒有,)(2xxf問(wèn))(xf能否在0 x可導(dǎo)?解解:由題設(shè))0(f00)0()(xfxfx0由夾逼準(zhǔn)那么0)0()(lim0 xfxfx0故)(xf在0 x可導(dǎo), 且0)0( f機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 2.3nRules for Finding Derivatives常數(shù)和根本初等函數(shù)的導(dǎo)數(shù) )(C0 )(x1x )(sin xxcos )(cosxxsin )(tan xx2sec )(cot xx2csc )(secxxxtan

12、sec )(cscxxxcotcsc )(xaaaxln )(xexe )(log xaaxln1 )(ln xx1 )(arcsin x211x )(arccosx211x )(arctanx211x )cot(arcx211x機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 求橢圓求橢圓191622yx在點(diǎn))3,2(23處的切線方程.解解: 橢圓方程兩邊對(duì)橢圓方程兩邊對(duì) x 求導(dǎo)求導(dǎo)8xyy920y2323xyyx1692323xy43故切線方程為323y43)2( x即03843 yx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 四那么運(yùn)算求導(dǎo)法那么四那么運(yùn)算求導(dǎo)法那么 定理定理.具有導(dǎo)數(shù)都在及函數(shù)xx

13、vvxuu)()()()(xvxu及的和、 差、 積、 商 (除分母為 0的點(diǎn)外) 都在點(diǎn) x 可導(dǎo), 且)()( )()() 1 (xvxuxvxu)()()()( )()()2(xvxuxvxuxvxu)()()()()()()()3(2xvxvxuxvxuxvxu下面分三部分加以證明,并同時(shí)給出相應(yīng)的推論和例題 .)0)(xv機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 此法那么可推行到恣意有限項(xiàng)的情形.證證: 設(shè), 那么vuvu )() 1 ()()()(xvxuxfhxfhxfxfh)()(lim)(0hxvxuhxvhxuh )()( )()(lim0hxuhxuh)()(lim0hxvhx

14、vh)()(lim0)()(xvxu故結(jié)論成立.wvuwvu)( ,例如機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例如,(2)vuvuvu )(證證: 設(shè)設(shè), )()()(xvxuxf那么有hxfhxfxfh)()(lim)(0hxvxuhxvhxuh)()()()(lim0故結(jié)論成立.)()()()(xvxuxvxuhhxuh )(lim0)(xu)(hxvhxv)( )(xu)(hxv推論推論: )() 1uC )()2wvuuC wvuwvuwvu )log()3xaaxlnlnaxln1機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 ( C為常數(shù) )()( lim0 xvhxvh)()()()()()(

15、xvhxvhxvxuxvhxuh)()(xvxu(3)2vvuvuvu證證: 設(shè)設(shè))(xf那么有hxfhxfxfh)()(lim)(0hh lim0,)()(xvxu)()(hxvhxu)()(xvxuhhxu )( )(xu)(xvhhxv )( )(xu)(xv故結(jié)論成立.)()()()()(2xvxvxuxvxu推論推論:2vvCvC機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 ( C為常數(shù) )例例. 解解:xsin41(21)1sin, )1sincos4(3xxxy.1xyy 及求 y)(xx)1sincos4(213xxx23( xx)1xy1cos4)1sin43( 1cos21sin27

16、27)1sincos4(3xx)1sincos4(3xx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 有限次四那么運(yùn)算的求導(dǎo)法那么 )(vuvu )( uCuC )( vuvuvuvu2vvuvu( C為常數(shù) )0( v機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 2.4nDerivatives of Trigonometric Functions Formula Formula.)(sin)(sin,sin)(4xxxxxf及求設(shè)函數(shù) 解解hxhxxhsin)sin(lim)(sin0 22sin)2cos(lim0hhhxh .cos x .cos)(sinxx即44cos)(sin xxxx.22 f(si

17、n x ) = cos xf(cos x) = - sin xnFind derivatives of other trig. functions using these derivatives and applying product rule and/or quotient rulexxxxxxfxxxxxxfxxx222222sec)(cos1)(cos)(sin)(cos)(tan)(cos)sin()(sin()cos()cos()(tan)cos()sin()tan( )(cscxxsin1x2sin)(sinxx2sin例例. 求證求證,sec)(tan2xx證證: .cotcs

18、c)(cscxxxxxxcossin)(tan x2cosxx cos)(sin)(cossinxx x2cosx2cosx2sinx2secxcosxxcotcsc類似可證:,csc)(cot2xx.tansec)(secxxx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 Derivatives of sec(x), csc(x) and cot(x)nAll are found by applying the product and/or quotient rules and using known derivatives of sin(x) and cos(x).xxfxxxfxxxf2csc)(

19、cotcotcsc)(csctansec)(sec2.5nThe Chain Rule復(fù)合函數(shù)求導(dǎo)法那么復(fù)合函數(shù)求導(dǎo)法那么For a composite function, its derivative is found by taking the derivative of the outer function, with respect to the inner function, times the derivative of the inner function with respect to x.nIf the composition consists of 3 or more fu

20、nctions, continue to take the derivative of the next inner function, with respect to the function within it, until, finally, the derivative is taken with respect to x.在點(diǎn) x 可導(dǎo), lim0 xxuxuuf)(xyxyx0limdd復(fù)合函數(shù)求導(dǎo)法那么復(fù)合函數(shù)求導(dǎo)法那么定理定理3.)(xgu )(ufy 在點(diǎn))(xgu 可導(dǎo)復(fù)合函數(shù) fy )(xg且)()(ddxgufxy在點(diǎn) x 可導(dǎo),證證:)(ufy 在點(diǎn) u 可導(dǎo), 故)

21、(lim0ufuyuuuufy)(當(dāng) 時(shí) )0u0故有)()(xgufuy)(uf)0()(xxuxuufxy機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 求以下函數(shù)的導(dǎo)數(shù)212sinxxy222222222212cos)1 (22)1 ()20(2)1 (212cos1212cosxxxxxxxxxxxxxxy例如,)(, )(, )(xvvuufyxydd)()()(xvufyuvxuyddvuddxvdd關(guān)鍵: 搞清復(fù)合函數(shù)構(gòu)造, 由外向內(nèi)逐層求導(dǎo).推行：此法那么可推行到多個(gè)中間變量的情形推行：此法那么可推行到多個(gè)中間變量的情形.機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 求解解: :,1111x

22、xxxy.y21222xxy12xx1 y1212x)2( x112xx例例.設(shè)),0( aaaxyxaaaxa解解: :1aaaxayaaaxln1axaaaxaln求.yaaxln機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 求解解:,1arctan2sin2xeyx.y1arctan) (2xy) (2sin xe2sin xe2cosxx221x1212xx2x21arctan2x2sin xe2cos x2sin xe112xx關(guān)鍵關(guān)鍵: 搞清復(fù)合函數(shù)構(gòu)造搞清復(fù)合函數(shù)構(gòu)造 由外向內(nèi)逐層求導(dǎo)由外向內(nèi)逐層求導(dǎo)機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 設(shè)設(shè)求,1111ln411arctan

23、21222xxxy.y解解: y22)1(1121x21xx) 11ln() 11ln(22xx111412x21xx1112x21xx2121xx221x21x231)2(1xxx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 Find the derivative (note this is the composition of 3 functions, therefore there will be 3 “pieces to the chain.)cotcsc3()csc(5)csccos()cscsin(2435353xxxxxxxyxxy3.7nHigher-Order Derivativesf

24、=2nd derivativef=3rd derivativef=4th derivative, etcnThe 2nd derivative is the derivative of the 1st derivative.nThe 3rd derivative is the derivative of the 2nd derivative, etc.定義定義.假設(shè)函數(shù))(xfy 的導(dǎo)數(shù))(xfy可導(dǎo),或,dd22xy即)( yy或)dd(dddd22xyxxy類似地 , 二階導(dǎo)數(shù)的導(dǎo)數(shù)稱為三階導(dǎo)數(shù) ,1n階導(dǎo)數(shù)的導(dǎo)數(shù)稱為 n 階導(dǎo)數(shù) ,y ,)4(y)(,ny或,dd33xy,dd44xyn

25、nxydd,)(xf的二階導(dǎo)數(shù) , 記作y )(xf 的導(dǎo)數(shù)為依次類推 ,分別記作那么稱機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 Velocity is the derivative of distance with respect to time (1st derivative) and Acceleration is the derivative of velocity with respect to time (2nd derivative of distance with respect to time)nUp (or right) is a positive velocity.nDown

26、(or left) is a negative velocity.nWhen an object reaches its peak, its velocity equals zero.)(tss 速度即sv加速度,ddtsv tvadd)dd(ddtst即)( sa引例：變速直線運(yùn)動(dòng)引例：變速直線運(yùn)動(dòng)機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 3.8nImplicit Differentiationn(An application of the chain rule!)ny is now considered as a function of x, therefore we apply the cha

27、in rule to ynApply all appropriate rules and solve for dy/dx.Find the derivativeyxyxxyydxdyxyyyxyxdxdydxdyyxyydxdyxxydxdyyydxdyxxyxyxy2222sec)cos()cos(2)cos(2)sec)cos(2)(sec)cos()cos(2)(sec) 1()cos(2)tan()sin(例例. 求橢圓求橢圓191622yx在點(diǎn))3,2(23處的切線方程.解解: 橢圓方程兩邊對(duì)橢圓方程兩邊對(duì) x 求導(dǎo)求導(dǎo)8xyy920y2323xyyx1692323xy43故切線方程

28、為323y43)2( x即03843 yx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 求求)0(sinxxyx的導(dǎo)數(shù) . 解解: 兩邊取對(duì)數(shù)兩邊取對(duì)數(shù) , 化為隱式化為隱式xxylnsinln兩邊對(duì) x 求導(dǎo)yy1xx lncos xxsin)sinlncos(sinxxxxxyx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 1) 對(duì)冪指函數(shù)vuy 可用對(duì)數(shù)求導(dǎo)法求導(dǎo) :uvylnlnyy1uv lnuvu)ln(uvuuvuyvvuuyvlnuuvv1闡明闡明: :按指數(shù)函數(shù)求導(dǎo)公式按冪函數(shù)求導(dǎo)公式留意留意:機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 2) 有些顯函數(shù)用對(duì)數(shù)求導(dǎo)法求導(dǎo)很方便 .例如例如,)1,

29、0,0(babaaxxbbaybax兩邊取對(duì)數(shù)yln兩邊對(duì) x 求導(dǎo)yybalnxaxb baxaxxbbaybalnxaxbbaxlnlnlnxbalnlnaxb機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 又如又如, )4)(3()2)(1(xxxxyuuu )ln(21lny對(duì) x 求導(dǎo)21yy)4)(3()2)(1(21xxxxy41312111xxxx兩邊取對(duì)數(shù)2ln1lnxx4ln3lnxx11x21x31x41x機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 設(shè))(xyy 由方程eyxey確定 , , )0(y解解: 方程兩邊對(duì) x 求導(dǎo), 得0yxyyey再求導(dǎo), 得2yey yxey)(02 y當(dāng)

30、0 x時(shí), 1y故由 得ey1)0(再代入 得21)0(ey 求. )0(y 機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 設(shè),)2(2)(sin32lntanxxxxxyxx求.y1y2y分別用對(duì)數(shù)微分法求.,21yy答案答案: :21yyy) 1sinln(sec)(sin2tanxxxx32ln)2(31xxxx)2(32)2(3ln21xxxxx機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 2.8nRelated RatesnA very, very important application of the derivative!nApplies to situations where more than

31、 one variable is changing with respect to time. nThe other variables are defined with respect to time, and we differentiate implicitly with respect to time. 相關(guān)變化率相關(guān)變化率)(, )(tyytxx為兩可導(dǎo)函數(shù)yx ,之間有聯(lián)絡(luò)tytxdd,dd之間也有聯(lián)絡(luò)稱為相關(guān)變化率相關(guān)變化率問(wèn)題解法:找出相關(guān)變量的關(guān)系式對(duì) t 求導(dǎo)得相關(guān)變化率之間的關(guān)系式求出未知的相關(guān)變化率機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 相關(guān)變化率相關(guān)變化率)(, )(tg

32、gtff為兩可導(dǎo)函數(shù)gf ,之間有聯(lián)絡(luò)tgtfdd,dd之間也有聯(lián)絡(luò)稱為相關(guān)變化率相關(guān)變化率問(wèn)題解法:找出相關(guān)變量的關(guān)系式對(duì) t 求導(dǎo)得相關(guān)變化率之間的關(guān)系式求出未知的相關(guān)變化率機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 一氣球從分開察看員一氣球從分開察看員500 m 處離地面鉛直上升處離地面鉛直上升,其速率為,minm140當(dāng)氣球高度為 500 m 時(shí), 察看員視野的仰角添加率是多少? 500h解解: 設(shè)氣球上升設(shè)氣球上升 t 分后其高度為分后其高度為h , 仰角為仰角為 ,那么tan500h兩邊對(duì) t 求導(dǎo)2sectddthdd5001知,minm140ddth h = 500m 時(shí),1t

33、an22tan1sec,2sec2td 0)minrad/(機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 由參數(shù)方程確定的函數(shù)的導(dǎo)數(shù)由參數(shù)方程確定的函數(shù)的導(dǎo)數(shù)假設(shè)參數(shù)方程)()(tytx可確定一個(gè) y 與 x 之間的函數(shù))(, )(tt可導(dǎo), 且,0 )( )(22tt那么0)( t時(shí), 有xyddxttyddddtxtydd1dd)()(tt0)( t時(shí), 有yxddyttxddddtytxdd1dd)()(tt(此時(shí)看成 x 是 y 的函數(shù) )關(guān)系,機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 例例. 拋射體運(yùn)動(dòng)軌跡的參數(shù)方程為拋射體運(yùn)動(dòng)軌跡的參數(shù)方程為 1tvx 求拋射體在時(shí)辰 t 的運(yùn)動(dòng)速度的大小和方向. 解解: 先求速度大小先求速度大小:速度的程度分量為,dd1vtx垂直分量為,dd2tgvty故拋射體速度大小22)dd()dd(tytxv2221)(gtvv再求速度方向(即軌跡的切線方向):設(shè) 為切線傾角,tanxyddtyddtxdd12vtgv 那么yxo2212tgtvy機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 前往 終了 拋射體軌跡的參數(shù)方程22121 tgtvytvx速度的程度分量,dd1vtx垂直分量,dd2tgvtytan12vt gv 在剛射出 (即 t = 0 )時(shí), 傾角為12arctanvv到達(dá)最高點(diǎn)的時(shí)辰,2gvt 高度ygv2221落地時(shí)辰,22gv