From e3263cf39fb0e52ae5571b7901cd11604dab08ee Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 31 May 2021 23:38:13 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=AC=AC=E4=BA=8C=E4=B8=89?= =?UTF-8?q?=E7=AB=A0=E4=B9=A0=E9=A2=98=E9=83=A8=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- advanced-math/exercise/1-limit/limit.pdf | Bin 170397 -> 177441 bytes advanced-math/exercise/1-limit/limit.tex | 275 +++++++----------- .../continuity-and-discontinuity.pdf | Bin 66530 -> 66494 bytes .../continuity-and-discontinuity.tex | 10 +- .../derivative-and-differentiate.pdf | Bin 104218 -> 127432 bytes .../derivative-and-differentiate.tex | 98 +++++-- advanced-math/knowledge/0-perpare/perpare.pdf | Bin 206761 -> 206736 bytes advanced-math/knowledge/0-perpare/perpare.tex | 6 +- .../function-and-limit.pdf | Bin 329238 -> 230347 bytes .../function-and-limit.tex | 26 +- .../derivatives-and-differential.pdf | Bin 143747 -> 142834 bytes ...derivatives-and-differential.synctex(busy) | 0 .../derivatives-and-differential.tex | 52 +--- ...heorem-and-applications-of-derivatives.pdf | Bin 219176 -> 147302 bytes ...heorem-and-applications-of-derivatives.tex | 73 +---- ...efinite-integral-and-definite-integral.pdf | Bin 244987 -> 166255 bytes ...efinite-integral-and-definite-integral.tex | 53 ++-- ...ral-calculus-of-multivariate-functions.pdf | Bin 0 -> 12845 bytes 18 files changed, 237 insertions(+), 356 deletions(-) delete mode 100644 advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.synctex(busy) create mode 100644 advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf diff --git a/advanced-math/exercise/1-limit/limit.pdf b/advanced-math/exercise/1-limit/limit.pdf index a8837992e631886826d21763168e3009c4d252b5..f9cd4ee4562861b8720cbce75062c113bb8a1972 100644 GIT binary patch delta 170647 zcmZU)Q*gHl@Y!BnLah)xnaL54{e*29sa6=$LPMw?_Xo3j70q#S~)Yb|Kcv%wt z{VjE)Kp-(knH;h3(K1ESLe-)edX1t=vi`BJ2mTzQP(o$v$jtsB^gXQV`4Q;rHTeGh zc|Kh|88vJmcB&+szBCi{pq6?>=`Mb+*-rozIGrVaLK*(}d{5uJ!Z$uvv+e$xm;!Yb ztcxp_e*#;988u#-tpfLNW)^Rc0@;}2pS&v0ejYzh!&)vuUkUFZ^V&M7BKOc<=82@4*I=FpI!Bty+UM6*=q?{)gxBs^Mn?Tx~s}vV0{7b zb=?hIT>kF!sdl+DYD-?2h5qXqRnQjZe;LAXWXhibHof0i-0VXMH=VhLt2lP!sw-sr0WptZ|#V3 zh(0zNU4EXH%uO9_EyBoNBTrk{Dt!GFk!Ki$VZY@F{nz_|%r1z}ZvzDwI0J?xA0eKK zBf1)a4l@@O{aa7{3{R}=fyYQ)U*iBZAVnVsY!v8F``)C}UXO@tCGH+`In zA7h~AX>({Hq{bG+B^Oe_Gqo}8uAZ}X((u3!8?UFcgAgI&l=L^WxW=0B)s-5t!7*qk z9l_Y)zy3=*p)K45g(L&0qLQ;JykA-c8 z1ZzdFCWk%;Aj#R_LSGZhb>2sSWsQq|Sr$6uAG<<(;9?JWRHjbAk+7dugciSW-E}KL z)x#P6ox(Q1fl~WZG#{2KL)d${eFAaZ^v6CaprU(!;}h{0IOrGS62{!Y?EjL)f0d(&4GcSX5;7Tpn}zv*(MQ&# z1FA5R_^aRWmr%CPb1_*eIEt7_#1X#9{{33D*4;1ego1@ro8k6I6bhEO25E)>S>kq4Eq(V=*L&G`J9J$Q)_Mz>z z$=eGs5r7WZuW?zhHh<*#=kp-OAL0fO{udMu!^oEzh5E(AzsR^tMTZ80&=WI2Cs#Tw zGcORW;>`_UsjbG3x;#Rgf4x5JNQ;I)tmLPA@-P8&UhNQc6zo>mN4Jf`Uv}(6iCvqH zG*~IZfmV(Z&1R5=$4F|f0j#|$i@o+9xt;eK4@gy@^41epi1))o$MZ>52N?jy)t8T(E2VkGs`=)u_M=89Z-Fmb}EM>B|rc3>8l9Ed8F?7Kf$wKcg*6r&Zhyu^!& zu5Y^FG$*OLW|y;a34$fd|Jvt6S-D4tDyp4(ECYcg!Q?wdw#0Fm2Im=2Lt8J2_Q8nD z1W>2pMSnXajd%W>u~jQ7yzoa>a22Qdt#_4=|}2~@a4Z- zXU2(@|5F_~JsF2P9bf3)ua<%5I%T%}QviKdG{rW)!S-SFG($@ONMU6(L=TP!ox>@WIaGDTps(9dahpAT_!Sxpy2g)ZXD=2t9r1WW-%P zoAvb3HG(r}lxWC#0&bG`l{e`FhSCvv>G)!T5Ob~J?l;6AC$;~E6fqZom5H0_e?oR! zcRZe~4Y}u&R)!NELYy_YOaCGSSU1oy{FLHcDNFk}23!kL`|&+sU~7!dg8mQ#3^zo= z-P{#=GWFhn4x zCK@Eoly3zgc3!c|izd>p1ELa_pR zR-X39rhIuJ(%=+Ba(t*gbJ7QkK6Z*AKXO%2jPJfshFx@P0!4jTdEd!A-5n)88*f~&3v4qG(e0B@w>&ujeRtZui+dkm{%4&cr48{Sp~hZ7 z5K|y3$Mfs`2zI)bb?Jr{7!i&ifkrLy$NRHx(}TPd5o~@p55(G=>EL~w+p!vZhhn}4a`+Lglh~=hR2HVDx9_;2)W`ZI8Hl(0pgM^@{Jvb|E-U1#% zsf+pDJef*%(#=1ku`!m3&LvAiyJMb^{h!{>nm4F}kmeCmSa;uf550n`^G)HnL$0|H z=Firz1qm{7tuQDNQg|cK8a9B{@PpYbV@4JcgjvPIWb?f&Vs6lj9~R<=JVE%?@w;}Su&AeMU*Q+r9KR@(y-|)48EA|K2YFRk&N=QG`f9)uL>v4vHmV*R zrEErw^LbHm(BIV{h!nSry5O_y&o>CtJUDlq^0Ox)oxNZaEFp2;i?3t)OjL*s^H?QS zr54n6Aim?E&ZZgKI1}~tRMJ7_4s6useG1f8CKsvPtok@x8t(-+4 z!8leZUZ)^r-TJ%KA$`DPxHv+e=vd{L1$Y!pE@~{PRO#%-C6Jizl2t<&f6R@zC{JFT zfG5A-by3Q*6#S|`2}o$P2hX60=+`0XRyp9DTO!@yaFX}wcHZKobedqU?+NW2_c2}z zR9SJUtLm+RYwbWUOq?bv;qU5`T0^LFTfynJ9We?@`)P2}PZz+QNv!f{(0z&XIvAvP zgmt~Q*#k6{NOtE&qIU&_QP_$!z#?9!20iWh(L%7T6k$rfVkmCu$4GBxn1$iQS%*k5 zjG1WnMDlNd>7nu(u6s5iv%LFb3My{#&s?Z`1PH6lHhSJ?a8aef6b&P{6aI_(;sTeP zPUcD}zUBc;TqB?|77{l8LZ){`D>5ngl;qje7P&yZ?3H_R9&(N{6pf{PW#66dFEyVZ zNf-+3k1u-ZsXCd{u0UxDRoo)=;U&>ieEAV?c?rKL=^BaSIsb6y4!E*A31&$BL;Kz< zp#WPlw=_dJjs%vVJ}NzQr|t(u7ngPZ!1x=P!bi=*R{-FN4N*ykG8U#;Z}pGPXHuy| z$-76%w@&|O&fg%9r91b@8LFI=bsmwLv6AV^af(E>#OAltvO*iw%4L*P1tmckIoBEJ zK1lo0N$pKq1zZw+KO~O|$ZcqRiU=K*X!BTbN~{uW6UI}7RP3c0y%Md`RISX4o8Z|g zw|v$V2q(a=1R=I~amZl8Na4|+s6CtAYmVfhEveR@!93K?wUD3?1slh|gsr{tr&d7aP&Aj z`#A-1b-hJ#J5}isS&a_!LaBBxBTM6WqeoF?USZG`Z1$pKyna_6YoYi!gC<#%rh+CW zC1P*1zlYEBSQ03=ummxJP<~N^?~6C3EW;B4;+YQ1er=W-kk?SaLzFlm+b0votW;5r z1DbWM8C{15gCtn>~>@T(VeU@N3-Qj5JGKZSo1O zInGZimYE3*u$y83Ot2N`j)(^9d2N7sw@4dtPLi}S?GsT#I~5>hP-(4; 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z>x+t{nWfY~*-}YVh)mpelP!1(NmOd9tSRpufo;pJp|LJNv@NY`(4rpHqFKGJ__cLG zv%RsN?@E2`M0D1`5|&4!<3l_Ba#W&?N}F&0O>PiRJ#=!b_K{E^T7?TkuZtYF$1)=v zkZl6f>t$9Cr~-Z0j*zye>xdk+Rju^x!NG_vkfX~JLBM~E7PYr!E-A(h*8)&(aZ1 zvK7vSP_*?cBS`BC)24UQp7*?K(vI>R=vi-qdDsD83w~ja*xbT-Bq#2x`Ui=!ZSacU zdWB2eo&QKrJc#i~C_ATdG>rtaQ^@+tB5aW~no@FEg(n%!_|o9ao_HB$5tOs@Ax%zc%D0STj>rRUcQi zk%pT-J8#u)azk~;vAdFyyD>oP&cP|NZ<7uq>J!V&fztn-FbJHkpQXG@s(b*yyx)Vc iYNLZ8_5+-^VS^$0u@2&s37VUgiyNMtTvT2R{{I7v0,b>0,c>0)$。\medskip -$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}$ +$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) -$=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}$ - -$=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) - -$=e^{\lim\limits_{x\to 0}\frac{\ln a+\ln b+\ln c}{1+1+1}}$ - -$=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。 +$=e^{\lim\limits_{x\to 0}\frac{\ln a+\ln b+\ln c}{1+1+1}}=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。 \textbf{例题:}求$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]$。 @@ -108,41 +102,23 @@ $=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。 综上: -$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]$ \medskip +$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]=\lim\limits_{n\to\infty}n\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}\right)$ \medskip -$=\lim\limits_{n\to\infty}n\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}\right)$ \medskip +$=\lim\limits_{n\to\infty}n\left[e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)\right]=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}n^2\left[\ln\left(1+\frac{1}{n}\right)-\dfrac{1}{n}\right]$ -$=\lim\limits_{n\to\infty}n\left[e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)\right]$ \medskip - -$=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}n^2\left[\ln\left(1+\frac{1}{n}\right)-\dfrac{1}{n}\right]$ - -$=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\dfrac{1}{2n^2}-\dfrac{1}{n}}{\dfrac{1}{n^2}}$ - -$=\dfrac{e^{\frac{1}{2}}}{2}\cdot\left(-\dfrac{1}{2}\right)$ - -$=-\dfrac{\sqrt{e}}{4}$ +$=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\dfrac{1}{2n^2}-\dfrac{1}{n}}{\dfrac{1}{n^2}}=\dfrac{e^{\frac{1}{2}}}{2}\cdot\left(-\dfrac{1}{2}\right)=-\dfrac{\sqrt{e}}{4}$ \subsection{三角函数关系式} \textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)$。\medskip -$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)$ \medskip +$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip -$= \lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip +$=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{x^4} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{x^2-\dfrac{1}{4}\sin^22x}{x^4}$ \medskip -$= \lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{x^4} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)$ \medskip +$=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{4}\cdot 2\sin 2x\cdot\cos 2x\cdot 2}{4x^3} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{2}\sin 4x}{4x^3}$ \medskip -$= \lim\limits_{x\to 0}\dfrac{x^2-\dfrac{1}{4}\sin^22x}{x^4}$ \medskip - -$= \lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{4}\cdot 2\sin 2x\cdot\cos 2x\cdot 2}{4x^3} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)$ \medskip - -$= \lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{2}\sin 4x}{4x^3}$ \medskip - -$= \lim\limits_{x\to 0}\dfrac{2-\dfrac{1}{2}\cos 4x\cdot 4}{12x^2}$ \medskip - -$= \dfrac{1}{6}\lim\limits_{x\to 0}\dfrac{1-\cos 4x}{x^2} (1-\cos x\sim \dfrac{1}{2}x^2)$ \medskip - -$= \dfrac{4}{3}$ +$=\lim\limits_{x\to 0}\dfrac{2-\dfrac{1}{2}\cos 4x\cdot 4}{12x^2}=\dfrac{1}{6}\lim\limits_{x\to 0}\dfrac{1-\cos 4x}{x^2} (1-\cos x\sim \dfrac{1}{2}x^2)=\dfrac{4}{3}$ \subsection{提取常数因子} @@ -178,25 +154,17 @@ $=\lim\limits_{t\to 0^-}\dfrac{\sqrt{t+4}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2}{ 幂次不高可以尝试洛必达:\medskip -$=\lim\limits_{t\to 0^-}\left(\dfrac{1}{2}\cdot\dfrac{\ln(2+t)}{\sqrt{t+4}}+\dfrac{\sqrt{t+4}}{2+t}\right)$\medskip - -$=-\left(\dfrac{1}{2}\cdot\dfrac{\ln 2}{2}+\dfrac{2}{2}\right)=-\dfrac{\ln 2}{4}-1$。 +$=\lim\limits_{t\to 0^-}\left(\dfrac{1}{2}\cdot\dfrac{\ln(2+t)}{\sqrt{t+4}}+\dfrac{\sqrt{t+4}}{2+t}\right)=-\left(\dfrac{1}{2}\cdot\dfrac{\ln 2}{2}+\dfrac{2}{2}\right)=-\dfrac{\ln 2}{4}-1$。 \subsection{幂指函数} 当出现$f(x)^{g(x)}$的类似幂函数与指数函数类型的式子,需要使用$u^v=e^{v\ln u}$。 -\textbf{例题:}求极限$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}$。\medskip +\textbf{例题:}求极限$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}$。 -$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}$ \medskip +$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln(x+\sqrt{1+x^2})}{x}} \left(\ln(x+\sqrt{1+x^2})'=\dfrac{1}{\sqrt{1+x^2}}\right)$\medskip -$=e^{\lim\limits_{x\to+\infty}\frac{\ln(x+\sqrt{1+x^2})}{x}} \left(\ln(x+\sqrt{1+x^2})'=\dfrac{1}{\sqrt{1+x^2}}\right)$\medskip - -$=e^{\lim\limits_{x\to+\infty}\frac{1}{\sqrt{1+x^2}}}$ - -$=e^0$ - -$=1$ +$=e^{\lim\limits_{x\to+\infty}\frac{1}{\sqrt{1+x^2}}}=e^0=1$ \subsection{有理化} @@ -214,17 +182,9 @@ $=1$ 又$\sqrt{x^2+100}$在$x\to-\infty$时本质为根号差,所以有理化: -$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)$ +$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)=\lim\limits_{x\to-\infty}x\dfrac{x^2+100-x^2}{\sqrt{x^2+100}-x}=\lim\limits_{x\to-\infty}\dfrac{100x}{\sqrt{x^2+100}-x}$\medskip -$=\lim\limits_{x\to-\infty}x\dfrac{x^2+100-x^2}{\sqrt{x^2+100}-x}$\medskip - -$=\lim\limits_{x\to-\infty}\dfrac{100x}{\sqrt{x^2+100}-x}$\medskip - -$\xRightarrow{\text{令}x=-t}\lim\limits_{t\to+\infty}\dfrac{-100t}{\sqrt{t^2+100}+t}$\medskip - -$=\lim\limits_{t\to+\infty}\dfrac{-100}{\sqrt{1+\dfrac{100}{t^2}}+1}$ - -$=-50$ +$\xRightarrow{\text{令}x=-t}\lim\limits_{t\to+\infty}\dfrac{-100t}{\sqrt{t^2+100}+t}=\lim\limits_{t\to+\infty}\dfrac{-100}{\sqrt{1+\dfrac{100}{t^2}}+1}=-50$\medskip \textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}$。 @@ -248,19 +208,9 @@ $=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{x\sin^2x}=\lim\limits_{x\to 0}\dfrac 又$x\to 0$,$\ln(1+x)\sim x$,所以$x\to 1$,$\ln x\sim x-1$: -$\lim\limits_{x\to 1^-}\ln x\ln(1-x)$ +$\lim\limits_{x\to 1^-}\ln x\ln(1-x)=\lim\limits_{x\to 1^-}(x-1)\ln(1-x)\xRightarrow{令t=1-x} =-\lim\limits_{t\to 0^+}t\ln t$ -$= \lim\limits_{x\to 1^-}(x-1)\ln(1-x)$ - -$\xRightarrow{令t=1-x} =-\lim\limits_{t\to 0^+}t\ln t$ - -$= -\lim\limits_{t\to 0^+}\dfrac{\ln t}{\dfrac{1}{t}}$ - -$= -\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{t}}{-\dfrac{1}{t^2}}$ - -$= \lim\limits_{t\to 0^+}t$ - -$= 0$ +$=-\lim\limits_{t\to 0^+}\dfrac{\ln t}{\dfrac{1}{t}}=-\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{t}}{-\dfrac{1}{t^2}}=\lim\limits_{t\to 0^+}t=0$ \subsection{倒代换} @@ -268,13 +218,9 @@ $= 0$ 当极限式子中含有分式中一般都需要用其倒数,把分式换成整式方便计算。 -\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$ +\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$ \medskip -$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$\medskip - -$= \lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}$\medskip - -$= \lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ +$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}=\lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ \medskip @@ -282,33 +228,19 @@ $= \lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2} 使用倒代换再洛必达降低幂次,令$t=\dfrac{1}{x^2}$ -$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$\medskip +$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{t\to+\infty}\dfrac{e^{-t}}{t^{-50}}=\lim\limits_{t\to+\infty}\dfrac{t^{50}}{e^t}=\lim\limits_{t\to+\infty}\dfrac{50t^{49}}{e^t}$ -$= \lim\limits_{t\to+\infty}\dfrac{e^{-t}}{t^{-50}}$\medskip +$=\cdots$ -$= \lim\limits_{t\to+\infty}\dfrac{t^{50}}{e^t}$\medskip - -$= \lim\limits_{t\to+\infty}\dfrac{50t^{49}}{e^t}$ - -$= \cdots$ - -$= \lim\limits_{t\to+\infty}\dfrac{50!}{e^t}$ - -$= 0$ +$=\lim\limits_{t\to+\infty}\dfrac{50!}{e^t}=0$ \textbf{例题:}求极限$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]$。 该式子含有分数,所以尝试使用倒数代换:\medskip -$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]$ \medskip +$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]\xRightarrow{\text{令}x=\frac{1}{t}}\lim\limits_{t\to 0^+}\left(\dfrac{e^t-1}{t^2}-\dfrac{1}{t}\right)=\lim\limits_{t\to 0^+}\dfrac{e^t-1-t}{t^2}$ -$\xRightarrow{\text{令}x=\frac{1}{t}}\lim\limits_{t\to 0^+}\left(\dfrac{e^t-1}{t^2}-\dfrac{1}{t}\right)$\medskip - -$\lim\limits_{t\to 0^+}\dfrac{e^t-1-t}{t^2}$ - -$\xRightarrow{\text{泰勒展开}e^t}\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{2}t^2}{t^2}$ - -$=\dfrac{1}{2}$ +$\xRightarrow{\text{泰勒展开}e^t}\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{2}t^2}{t^2}=\dfrac{1}{2}$ \subsubsection{\texorpdfstring{$\infty-\infty$}\ 型} @@ -330,21 +262,11 @@ $=\dfrac{1}{6}\lim\limits_{n\to\infty}\dfrac{n+1}{n}\times\dfrac{n+2}{n}\times\c 这里可以使用等价无穷小$e^x-1\sim x$。 -$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}$ +$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}=e^{\lim\limits_{x\to 0}\frac{e}{x}\ln\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)}$ -$=e^{\lim\limits_{x\to 0}\frac{e}{x}\ln\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)}$ +$=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}-1\right)}=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}-n}{n}\right)}$ -$=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}-1\right)}$ - -$=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}-n}{n}\right)}$ - -$=e^{\frac{e}{n}\lim\limits_{x\to 0}\left(\frac{e^x-1}{x}+\frac{e^{2x}-1}{x}+\cdots+\frac{e^{nx}-1}{x}\right)}$ - -$=e^{\frac{e}{n}[1+2+\cdots+n]}$ - -$=e^{\frac{e}{n}\cdot\frac{n(1+n)}{2}}$ - -$=e^{\frac{e(1+n)}{2}}$ +$=e^{\frac{e}{n}\lim\limits_{x\to 0}\left(\frac{e^x-1}{x}+\frac{e^{2x}-1}{x}+\cdots+\frac{e^{nx}-1}{x}\right)}=e^{\frac{e}{n}[1+2+\cdots+n]}=e^{\frac{e}{n}\cdot\frac{n(1+n)}{2}}=e^{\frac{e(1+n)}{2}}$ \textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$\medskip @@ -360,9 +282,7 @@ $=-1+\lim\limits_{x\to 0}\dfrac{(1-\sqrt{\cos 2x})+\sqrt{\cos 2x}-\sqrt{\cos 2x} $=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(\cos 2x-1)+\sqrt{\cos 2x}(1-\sqrt[3]{\cos 3x})}{-\dfrac{x^2}{2}}$ -$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(-\dfrac{4x^2}{2})+\left(-\dfrac{1}{3}\right)\left(-\dfrac{9x^2}{2}\right)}{-\dfrac{x^2}{2}}$ - -$=-6$ +$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(-\dfrac{4x^2}{2})+\left(-\dfrac{1}{3}\right)\left(-\dfrac{9x^2}{2}\right)}{-\dfrac{x^2}{2}}=-6$ \section{基本计算方式} @@ -376,15 +296,9 @@ $=-6$ 重要极限有两个,但是$\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$这个很少用,因为往往用等价无穷小替代了,而$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=e$则用的较多,当出现分数幂的幂指函数时,不要先去取对数,而是使用重要极限看看能不能转换。\medskip -\textbf{例题:}求$\lim\limits_{x\to\infty}\left(\dfrac{3+x}{6+x}\right)^{\frac{x-1}{2}}$。 +\textbf{例题:}求$\lim\limits_{x\to\infty}\left(\dfrac{3+x}{6+x}\right)^{\frac{x-1}{2}}$。\medskip -$=\lim\limits_{x\to\infty}\left(1-\dfrac{3}{6+3}\right)^{\frac{6+x}{-3}\cdot\frac{-3}{6+x}\cdot\frac{x-1}{2}}$ - -$=\lim\limits_{x\to\infty}e^{\frac{-3}{6+x}\cdot\frac{x-1}{2}}$ - -$=\lim\limits_{x\to\infty}e^{-\frac{3}{2}\cdot\frac{x-1}{x+6}}$ - -$=e^{-\frac{3}{2}}$。 +$=\lim\limits_{x\to\infty}\left(1-\dfrac{3}{6+3}\right)^{\frac{6+x}{-3}\cdot\frac{-3}{6+x}\cdot\frac{x-1}{2}}=\lim\limits_{x\to\infty}e^{\frac{-3}{6+x}\cdot\frac{x-1}{2}}=\lim\limits_{x\to\infty}e^{-\frac{3}{2}\cdot\frac{x-1}{x+6}}=e^{-\frac{3}{2}}$。\medskip \textbf{例题:}求$\lim\limits_{x\to\infty}\left(\dfrac{2x+3}{2x+1}\right)^{x+1}$。\medskip @@ -422,7 +336,45 @@ $\therefore \lim\limits_{x\to 0}x\left[\dfrac{10}{x}\right]=10$。 \subsection{拉格朗日中值定理} -对于形如$f(a)-f(b)$的极限式子就可以使用拉格朗日中值定理,这个$f(x)$为任意的函数。 +对于形如$f(a)-f(b)$的极限式子就可以使用拉格朗日中值定理,这个$f(x)$为任意的函数。使用拉格朗日中值定理最重要的还是找到这个$f(x)$。 + +\subsubsection{证明不等式} + +证明不等式最重要的还是找到$f(x)$,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换。 + +\paragraph{对数函数特性} \leavevmode \medskip + +对于对数函数,要记住其特定的性质:$\log_n(\dfrac{a}{b})=\log_na-\log_nb$。 + +\textbf{例题:}设$a>b>0$,证明:$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$。 + +因为$\ln\dfrac{a}{b}=\ln a-\ln b$,所以令$f(x)=\ln x$。 + +所以根据拉格朗日中值定理:$\ln a-\ln b=f'(\xi)(a-b)$($\xi\in(b,a)$)。 + +又$f'(\xi)=\dfrac{1}{\xi}$,所以$\ln a-\ln b=\dfrac{a-b}{\xi}$。 + +又$\xi\in(b,a)$,所以$\dfrac{1}{\xi}\in(\dfrac{1}{a},\dfrac{1}{b})$。 + +所以$\dfrac{a-b}{a}<\dfrac{a-b}{\xi}<\dfrac{a-b}{b}$,从而$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$,得证。 + +\paragraph{查找特定值} \leavevmode \medskip + +对于证明一种不等式,如果里面没有差式,也无法转换为差式,那么就可以考虑制造差式,对于$f(x)$一般选择更高阶的,$a$选择$x$,$b$要根据题目和不等式设置一个常数。 + +一般是0或1。可以先尝试1。 + +\textbf{例题:}当$x>1$时,证明$e^x>ex$。 + +题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 + +然后选择一个常数作为$b$值,可以先选一个1作为$b$值:$f(x)-f(1)=f'(\xi)(x-1)$。 + +从而$e^x-e=e^\xi(x-1)$,$\xi\in(1,x)$,所以$e^x-e>e(x-1)$,即$e^x>ex$,得证。 + +\subsubsection{极限计算} + +可以将极限式子中形如$f(a)-f(b)$的极限部分使用拉格朗日中值定理进行替换。 \textbf{例题:}求极限$\lim\limits_{n\to\infty}n^2\left(\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\right)$。\medskip @@ -493,17 +445,9 @@ $\therefore \text{原式}=\dfrac{\dfrac{1}{x}x^3+o(x^3)}{-\dfrac{1}{2}x^3+o(x^3) 令目标$\dfrac{f(x)-1}{x}=t$,$\therefore f(x)=tx+1$。\medskip -$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}$\medskip +$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=\lim\limits_{x\to 0}\dfrac{\ln(1-x)+tx^2+x}{x^2} (\text{泰勒展开})$\medskip -$=\lim\limits_{x\to 0}\dfrac{\ln(1-x)+tx^2+x}{x^2} (\text{泰勒展开})$\medskip - -$=\lim\limits_{x\to 0}\dfrac{-x-\dfrac{x^2}{2}+tx^2+x}{x^2}$\medskip - -$=\lim\limits_{x\to 0}\dfrac{\left(t-\dfrac{1}{2}\right)x^2}{x^2}$\medskip - -$=\lim\limits_{x\to 0}\left(t-\dfrac{1}{2}\right)$ - -$=0$ +$=\lim\limits_{x\to 0}\dfrac{-x-\dfrac{x^2}{2}+tx^2+x}{x^2}=\lim\limits_{x\to 0}\dfrac{\left(t-\dfrac{1}{2}\right)x^2}{x^2}=\lim\limits_{x\to 0}\left(t-\dfrac{1}{2}\right)=0$ $\therefore\lim\limits_{x\to 0}t=\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}=\dfrac{1}{2}$。 @@ -583,15 +527,9 @@ $\because\lim\limits_{x\to 0^-}\dfrac{\sin x}{\ln(1+3x)}=\lim\limits_{x\to 0^-}\ $\therefore a=-1$,从而让极限式子变为一个商的形式:\medskip -$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}$\medskip +$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x-1}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{x}$\medskip -$=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x-1}$\medskip - -$=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{x}$\medskip - -$=\lim\limits_{x\to 0^+}b\cos x-1$\medskip - -$=b-1=\dfrac{1}{3}$\medskip +$=\lim\limits_{x\to 0^+}b\cos x-1=b-1=\dfrac{1}{3}$\medskip $\therefore a=-1,b=\dfrac{4}{3}$。 @@ -607,19 +545,13 @@ $\therefore a=-1,b=\dfrac{4}{3}$。 $\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$ -$= \lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}\right)+\lim\limits_{x\to 0^+}\left(a\cdot\arctan\dfrac{1}{x}\right)$ +$=\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}\right)+\lim\limits_{x\to 0^+}\left(a\cdot\arctan\dfrac{1}{x}\right)$ -$= \lim\limits_{x\to 0^+}\left(\dfrac{0\cdot\left(e^{\frac{2}{x}}\right)^2+e^{\frac{1}{x}}-\pi}{1\cdot\left(e^{\frac{2}{x}}\right)^2+1}\right)+a\cdot\dfrac{\pi}{2}$ - -$= a\cdot\dfrac{\pi}{2}$ +$=\lim\limits_{x\to 0^+}\left(\dfrac{0\cdot\left(e^{\frac{2}{x}}\right)^2+e^{\frac{1}{x}}-\pi}{1\cdot\left(e^{\frac{2}{x}}\right)^2+1}\right)+a\cdot\dfrac{\pi}{2}=a\cdot\dfrac{\pi}{2}$ \medskip -$\lim\limits_{x\to 0^-}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$ - -$= -\pi+a\cdot\left(-\dfrac{\pi}{2}\right)$ - -$= -\pi-\dfrac{\pi}{2}\cdot a$ +$\lim\limits_{x\to 0^-}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)=-\pi+a\cdot\left(-\dfrac{\pi}{2}\right)=-\pi-\dfrac{\pi}{2}\cdot a$ 因为极限值具有唯一性,所以$-\pi-\dfrac{\pi}{2}a=\dfrac{\pi}{2}a$,所以$a=-1$,极限值为$-\dfrac{\pi}{2}$。 @@ -627,21 +559,36 @@ $= -\pi-\dfrac{\pi}{2}\cdot a$ 函数的连续性代表:极限值=函数值。 +\subsubsection{判断函数连续} + +题目给出函数,往往是分段函数,然后判断分段点的连续性。\medskip + +\textbf{例题:}讨论函数$f(x)=\left\{\begin{array}{lcl} + \left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}},& & x>0 \\ + e^{-\frac{1}{2}}, & & x\leqslant 0 +\end{array}\right.$在$x=0$处的连续性。 + +因为$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}}=e^{\lim\limits_{x\to 0^+}\frac{1}{x}\ln[\frac{(1+x)^{\frac{1}{x}}}{e}]}$。 + +又$\lim\limits_{x\to 0^+}\dfrac{1}{x}\ln\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]=\lim\limits_{x\to 0^+}\dfrac{1}{x}\left[\dfrac{1}{x}\ln(1+x)-1\right]=\lim\limits_{x\to 0^+}\dfrac{\ln(1+x)-x}{x^2}$ + +$=\lim\limits_{x\to 0^+}\dfrac{\dfrac{1}{1+x}-1}{2x}=\lim\limits_{x\to 0^+}-\dfrac{1}{2(1+x)}=-\dfrac{1}{2}$。 + +$\therefore\lim\limits_{x\to 0^+}f(x)=e^{-\frac{1}{2}}$。 + +又$\lim\limits_{x\to 0^-}f(x)=\lim\limits_{x\to 0^-}e^{-\frac{1}{2}}=e^{-\frac{1}{2}}$,且$f(0)=e^{-\frac{1}{2}}$。 + +从而$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^-}f(x)=f(0)$,所以$f(x)$在$x=0$处连续。 + +\subsubsection{连续性求极限} + \textbf{例题:}函数在$f(x)$在$x=1$处连续,且$f(1)=1$,求$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$。 根据题目,所求的$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$中,唯一未知的且会随着$x\to+\infty$而变换就是$f\left(x^{\frac{1}{x}}\right)$。如果我们可以求出这个值就可以了。 而我们对于$f(x)$的具体的关系是未知的,只知道$f(1)=1$。那么先需要考察$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}$的整数最大值。 -$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}$ - -$=e^{\lim\limits_{x\to+\infty}\frac{\ln x}{x}}$ - -$=e^{\lim\limits_{x\to+\infty}\frac{1}{x}}$ - -$=e^0$ - -$=1$ +$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln x}{x}}=e^{\lim\limits_{x\to+\infty}\frac{1}{x}}=e^0=1$ $\therefore\lim\limits_{x\to+\infty}f(x^{\frac{1}{x}})=f(1)=1$。 @@ -655,29 +602,19 @@ $\therefore\lim\limits_{x\to+\infty}f(x^{\frac{1}{x}})=f(1)=1$。 首先看题目,给出的递推式设计到二阶递推,即存在三个数列变量,所以我们必须先求出对应的数列表达式。因为这个表达式涉及三个变量,所以尝试对其进行变型: -$a_{n+1}-a_n$ - -$=\dfrac{a_{n-1}-a_n}{2}$ - -$=\left(-\dfrac{1}{2}\right)(a_n-a_{n-1})$ - -$=\left(-\dfrac{1}{2}\right)^2(a_{n-1}-a_{n-2})$ +$a_{n+1}-a_n=\dfrac{a_{n-1}-a_n}{2}=\left(-\dfrac{1}{2}\right)(a_n-a_{n-1})=\left(-\dfrac{1}{2}\right)^2(a_{n-1}-a_{n-2})$ $=\cdots$ -$=\left(-\dfrac{1}{2}\right)^n(a_1-a_0)$ - -$= \left(-\dfrac{1}{2}\right)^n$ +$=\left(-\dfrac{1}{2}\right)^n(a_1-a_0)=\left(-\dfrac{1}{2}\right)^n$ 然后得到了$a_{n+1}-a_n=\left(-\dfrac{1}{2}\right)^n$,而需要求极限,所以使用列项相消法的逆运算: -$a_n= (a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots+(a_1-a_0)+a_0$\medskip +$a_n=(a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots+(a_1-a_0)+a_0$\medskip -$= \left(-\dfrac{1}{2}\right)^{n-1} + \left(-\dfrac{1}{2}\right)^{n-2} + \cdots + \left(-\dfrac{1}{2}\right)^0$\medskip +$=\left(-\dfrac{1}{2}\right)^{n-1} + \left(-\dfrac{1}{2}\right)^{n-2} + \cdots + \left(-\dfrac{1}{2}\right)^0$\medskip -$= \dfrac{1\cdot\left(1-\left(-\dfrac{1}{2}\right)^n\right)}{1-\left(-\dfrac{1}{2}\right)}$\medskip - -$= \dfrac{2}{3}\left[1-\left(-\dfrac{1}{2}\right)^n\right]$\medskip 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zgtVnk?1)dYEkS22*Ijqgt%!eLK`072EIqdP#47jJnkoWo)F&~w(BDIi^u9+V7{{Yw80@nZl diff --git a/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.tex b/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.tex index abfdbf5..6b69262 100644 --- a/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.tex +++ b/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.tex @@ -91,15 +91,11 @@ $f(x)$在$R$上连续。 $\lim\limits_{x\to 0^+}\dfrac{e^{ax^3}-1}{x-\arcsin x}=6$\medskip -$\therefore\lim\limits_{x\to 0^+}\dfrac{e^{ax^3}-1}{x-\arcsin x}$\medskip +$\therefore\lim\limits_{x\to 0^+}\dfrac{e^{ax^3}-1}{x-\arcsin x}=\lim\limits_{x\to 0^+}\dfrac{ax^3}{x-\arcsin x}$\medskip -$=\lim\limits_{x\to 0^+}\dfrac{ax^3}{x-\arcsin x}$\medskip +$\text{令}t=\arcsin x\Rightarrow=\lim\limits_{x\to 0^+}\dfrac{a\sin^3t}{\sin t-t}=a\lim\limits_{x\to 0^+}\dfrac{t^3}{\sin t-t}=a\lim\limits_{x\to 0^+}\dfrac{3t^2}{\cos t-1}$ -$\text{令}t=\arcsin x\Rightarrow=\lim\limits_{x\to 0^+}\dfrac{a\sin^3t}{\sin t-t}$ - -$=a\lim\limits_{x\to 0^+}\dfrac{t^3}{\sin t-t}$\medskip - -$=a\lim\limits_{x\to 0^+}\dfrac{3t^2}{\cos t-1}=-6a=6$。\medskip +$=-6a=6$。 $\therefore a=-1$时$f(x)$在$R$上连续。\medskip diff --git a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.pdf b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.pdf index 1faab28588c99184d4a4968fbefef73f52e37656..42d99d196e9d1b4590bfcd53607033285c5347ce 100644 GIT binary patch delta 122752 zcmZs?V{k4^)UFxZwr$(CjUC%|p6uARZQI$gZQHh!d1ub4srl;6zwSR>)mW>qzSrHE zwGd;6kjUgpViNSs4D2xEM|an6Fs!Tz$m9T4HUR+`7guL9BRd$+jq42AxGfI2(c5pd zACRgCYGt8#(Ev6f2k;k+qrKImWYNCglBa45mWS@6^>gBy2BvEKbc&i^MKUu^0Owl3 zuTfIDG&AI+{O=&(?O^ov{muE|_xQ1F_2gBso*0Tbk)y5)0yWHbsrxOXG%vJ{Fdk4o zS__Uupd5xAeK_h+2k4~dr20#~u*1<=!Olq1pZ-z`#MX)b?sa+!9Nkq?68hZlDK 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z;CI$N(N|%YUgf+w*3bLD`1(pXQKA?aqmXb%B)zM!!}Y%T8Su>*Q*&;C@L(BPSUEYN L$;d?I#i0KW&uics diff --git a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex index 8c6b876..4873b3a 100644 --- a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex +++ b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex @@ -35,7 +35,7 @@ \pagestyle{plain} \setcounter{page}{1} \section{一阶导数} -\subsection{幂指函数求导} +\subsection{幂指函数导数} 形如$f(x)^{g(x)}$的幂指函数求导也可以类似幂指函数的求极限方法。既可以取$e$为底的指数也可以取对数。 @@ -117,15 +117,11 @@ $f'(x)=\dfrac{\dfrac{-x}{1-x}-\ln(1-x)}{x^2}=\dfrac{x-(x-1)\ln(1-x)}{x^2(x-1)}\, 然后需要求分段点$x=0$处的导数。 -可以由导数的定义:、 +可以由导数的定义: 根据导数的定义是某点偏移量的极限值$\lim\limits_{x\to x_0}\dfrac{f(x)-f(x_0)}{x-x_0}$: -$f'(0)=\lim\limits_{x\to 0}\dfrac{f(x)-f(0)}{x-0}$\medskip - -$=\dfrac{\dfrac{\ln(1-x)}{x}-(-1)}{x-0}$\medskip - -$=\dfrac{\dfrac{\ln(1-x)}{x}+1}{x}$\medskip +$f'(0)=\lim\limits_{x\to 0}\dfrac{f(x)-f(0)}{x-0}=\dfrac{\dfrac{\ln(1-x)}{x}-(-1)}{x-0}=\dfrac{\dfrac{\ln(1-x)}{x}+1}{x}$\medskip $=\dfrac{\ln(1-x)+x}{x^2}$ @@ -169,7 +165,49 @@ $\therefore\lim\limits_{x\to 0^+}x^{\alpha-2}=0$。 $\therefore\alpha-2>0$,从而$\alpha>2$。 -\subsection{已知导数求极限} +\section{极限与导数} + +导数的定义由极限产生,所以其之间是可以互相转换的,当求一个导数时可以寻找是否能求出其极限。 + +\subsection{极限求导数} + +在导数这一章中的极限不会直接给出极限,而是会给出导数或函数的相关定义,来求极限,再根据导数定义转换为导数。同时要注意这里不止会有导数定义,还会有函数等性质。 + +\textbf{例题:}已知$f(x)$是周期为5的连续函数,它在$x=0$的某个邻域内满足关系式:$f(1+\sin x)-3f(1-\sin x)= 8x+o(x)$,且$f(x)$在$x=1$处可导,求曲线$y=f(x)$在点$(6,f(6))$处的切线方程。 + +因为这是个函数等式,而我们最后要求的是一个导数,所以先尝试对其直接求极限,令$x\to0$: + +$f(1)-3f(1)=0$,从而得到了一个函数值$f(1)=0$。 + +然后再对这个式子思考,等式右边为$8x$,而除以$x$就变成了8,而再对其求极限,右边就彻底变成了一个常数8: + +$\lim\limits_{x\to 0}\dfrac{f(1+\sin x)-3f(1-\sin x)}{x}=8$。对式子左边进行变形: + +$\lim\limits_{x\to 0}\dfrac{f(1+\sin x)-3f(1-\sin x)}{x}=\lim\limits_{x\to 0}\dfrac{f(1+\sin x)-3f(1-\sin x)}{\sin x}$。 + +令$t=\sin x$: + +$=\lim\limits_{t\to 0}\dfrac{f(1+t)-3f(1-t)}{t}=\lim\limits_{t\to 0}\dfrac{f(1+t)-f(1)}{t}+3\lim\limits_{t\to 0}\dfrac{f(1-t)-f(1)}{-t}$。 + +因为$t\to 0$,所以$1-t$和$1+t$都是$t=1$时的导数$f'(x)$的定义:$=4f'(1)=8$,从而$f'(1)=2$。 + +由$f(x)$的周期为5,所以$f(6)=f(1)=0$,$f'(6)=f'(1)=2$,所以曲线$y=f(x)$在$(6,f(6))$即$(6,0)$处的切线方程为$y-0=2(x-6)$即$2x-y-12=0$。 + +15.当正在高度H水平飞行的飞机开始向机场跑道下降时,如图2-16所示从飞机到北场的水平地面距离为L.假设飞机下降的路径为三次函 + +数y=ax' +bx2 +cx+d的图形,其中yl..=H.y1..o=0.试 + +确定飞机的降落路径. + +16.甲船以6 km/h的速率向东行驶,乙船以8 km/h + +的速事向南行驶在中午十二点整,乙船位于甲船之北 + +L01 + +16 km处问下午一点整两船相离的速率为多少? + +\subsection{导数求极限} 题目会给出对应的导数以及相关条件,并要求求一个极限,这个极限式子并不是个随机的式子,而一个是与导数定义相关的极限式子,所需要的就是将极限式子转换为导数定义的相关式子。 @@ -183,9 +221,7 @@ $\therefore\alpha-2>0$,从而$\alpha>2$。 然后需要转换目标的极限式子,因为目标式子倒过来的式子类似于导数定义的$f'(x)=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$结构。所以我们可以先求其倒数式子:\medskip -$=\lim\limits_{h\to 0}\dfrac{f(5-2\sin h)-f(5)}{h}$ - -$=\lim\limits_{h\to 0}\dfrac{f(5-2\sin h)-f(5)}{-2\sin h}\cdot\dfrac{-2\sin h}{h}$ +$=\lim\limits_{h\to 0}\dfrac{f(5-2\sin h)-f(5)}{h}=\lim\limits_{h\to 0}\dfrac{f(5-2\sin h)-f(5)}{-2\sin h}\cdot\dfrac{-2\sin h}{h}$ $=-2f'(5)=-2\times 1=-2$ @@ -199,17 +235,9 @@ $\therefore\lim\limits_{h\to 0}\dfrac{h}{f(5-2\sin h)-f(5)}=-\dfrac{1}{2}$。 设$\dfrac{1}{n}=x$,则: -$=\lim\limits_{x\to 0}(f(x))^{\frac{x}{1-\cos x}}$ +$=\lim\limits_{x\to 0}(f(x))^{\frac{x}{1-\cos x}}=e^{\lim\limits_{x\to 0}\frac{x}{1-\cos x}\ln f(x)}=e^{2\lim\limits_{x\to 0}\frac{\ln f(x)}{x}}=e^{2\lim\limits_{x\to 0}\frac{f(x)-1}{x}}$ -$=e^{\lim\limits_{x\to 0}\frac{x}{1-\cos x}\ln f(x)}$ - -$=e^{2\lim\limits_{x\to 0}\frac{\ln f(x)}{x}}$ - -$=e^{2\lim\limits_{x\to 0}\frac{f(x)-1}{x}}$ - -$=e^{2\lim\limits_{x\to 0}\frac{f(x)-f(0)}{x-0}}$ - -$=e^{2f'(0)}=e^6$。 +$=e^{2\lim\limits_{x\to 0}\frac{f(x)-f(0)}{x-0}}=e^{2f'(0)}=e^6$。 \section{高阶导数} @@ -257,12 +285,22 @@ $\quad\quad\quad+x^2(x+1)^2(x+2)^2\cdots 2(x+n)$ 因为原函数的一阶导数是$\dfrac{\textrm{d}y}{\textrm{d}x}$,而反函数就是对原函数的$xy$对调,所以其反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}$。 +当求反函数的高阶导数时需要将分子分母同时除以$\textrm{d}x$。 + \textbf{例题:}已知$y=x+e^x$,求其反函数的二阶导数。 -$y=x+e^x$的反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{1+e^x}$。 +$y=x+e^x$的反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{1+e^x}$。\medskip 所以二阶导数为$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}=\dfrac{\textrm{d}\left(\dfrac{1}{1+e^{x}}\right)}{\textrm{d}y}=\dfrac{\dfrac{\textrm{d}\left(\dfrac{1}{1+e^{x}}\right)}{\textrm{d}x}}{\dfrac{\textrm{d}y}{\textrm{d}x}}=-\dfrac{e^x}{(1+e^x)^3}$。 +\textbf{例题:}已知$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{y'}$,求$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}$和$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}$。 + +其实就是求$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}$和$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}$,$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{y'}$这个条件只是让我们用$y'$来表示结果而已。 + +$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}=\dfrac{\textrm{d}\dfrac{\textrm{d}x}{\textrm{d}y}}{\textrm{d}y}=\dfrac{\dfrac{\textrm{d}\dfrac{\textrm{d}x}{\textrm{d}y}}{\textrm{d}x}}{\dfrac{\textrm{d}y}{\textrm{d}x}}=\dfrac{-\dfrac{y''}{(y')^2}}{y'}=-\dfrac{y''}{(y')^3}$。\medskip + +$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}=\dfrac{\textrm{d}\dfrac{\textrm{d}^2x}{\textrm{d}^2y}}{\textrm{d}y}=\dfrac{\dfrac{\textrm{d}\dfrac{\textrm{d}^2x}{\textrm{d}^2y}}{\textrm{d}x}}{\dfrac{\textrm{d}y}{\textrm{d}x}}=\dfrac{-\dfrac{y'''y'-3(y'')^2}{(y')^4}}{y'}=\dfrac{3(y'')^2-y'y'''}{(y')^5}$。 + \section{微分} \section{隐函数与参数方程} @@ -271,6 +309,22 @@ $y=x+e^x$的反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{ $\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\dfrac{\textrm{d}y}{\textrm{d}t}}{\dfrac{\textrm{d}x}{\textrm{d}t}}$。 +\subsection{隐函数应用} + +对于隐函数的应用题,最重要的是找到等价于$\textrm{d}x$和$\textrm{d}y$的两个相关变量。最简单的就是根据题目最后面的所求问题所涉及的变量设置其为变量和因变量。 + +\textbf{例题:}落在平静水面上的石头,产生同心波纹。若最外一圈波半径的增大速率总是6m/s,问在2s末扰动水面面积增大的速率为多少? + +首先根据题目最后的要求的是面积,所以肯定要设一个面积变量,随时间变动而改变,所以也一定会设一个时间变量,同时还给出一个条件是半径增大速度,所以也会有一个半径的变量。同时要求的是面积增大速率,正好跟另外两个变量相关,时间跟半径和面积都相关,所以时间就是中间变量。 + +从而设最外一圈波的半径为$r=r(t)$,圆的面积$S=S(t)$。根据$S$和$r$的公式$S=\pi r^2$,因为求的是随时间变化的速率,所以其两端分别对$t$求导,得: + +$\dfrac{\textrm{d}S}{\textrm{d}t}=2\pi r\dfrac{\textrm{d}r}{\textrm{d}t}$。当$t=2$时,$r=6\times2=12$,代入上式得:\medskip + +$\dfrac{\textrm{d}S}{\textrm{d}t}\bigg|_{t=2}=2\pi\cdot12\cdot6=144\pi$。 + +\subsection{分段参数方程} + \textbf{例题:}已知$y=y(x)$由参数方程$\left\{\begin{array}{lcl} x=\dfrac{1}{2}\ln(1+t^2) \\ y=\arctan t diff --git a/advanced-math/knowledge/0-perpare/perpare.pdf b/advanced-math/knowledge/0-perpare/perpare.pdf index deac789b81b972e4df38a4c9eea42e538122b148..b711390bf65e0abb9fa12ae57dc9af99e05c2c5a 100644 GIT binary patch delta 39892 zcmZU3Q*hu-@NKfOZQHhOTN~ROJHMD4+qS*&#~`9tLcV-#}d z9vYURE;>Uzl9#!`rP8~+l8U+!oR@zv@gs$1Lj34) z!=*7!_PPB7OcpR6>li+}P1aelbot_ha9Z4d`b=bL0RYtDoyI z2f^oGaQ+md0Y6=l*E6owq;z3U9{%=r^tubhw6>upqZ=6KM)vD=ONC=vWHy%&CW@k= 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zkWG7RNJZPIfFq92*nzE|3;V{~f7C+?#|*2u=Z=n5 zJ3EdiG%Q11Z1e?%XOA#Q5}XdMYbuOR-A(3W#{Sk_=c?HW2OPgD#k#U`4iSSw$~#Dw g30$。 +\textbf{例题:}证明$\dfrac{1}{x+1}<\ln(1+\dfrac{1}{x})<\dfrac{1}{x}$,其中$x>0$。 首先因为证明中间项无法进行直接处理,又看到是一个对数,所以进行通分:$\ln(1+\dfrac{1}{x})=\ln\dfrac{x+1}{x}=\ln(x+1)-\ln x$。 @@ -993,7 +993,7 @@ $a^\alpha\cdot a^\beta=a^{\alpha+\beta},\dfrac{a^\alpha}{a^\beta}=a^{\alpha-\bet 以后的华里士公式(点火公式)会使用到,如下面的题目: -\textbf{例题5:}计算$\int_0^{\frac{\pi}{2}}\sin^{10}x\textrm{d}x$与$\int_0^{\frac{\pi}{2}}\cos^9x\textrm{d}x$。\medskip +\textbf{例题:}计算$\int_0^{\frac{\pi}{2}}\sin^{10}x\textrm{d}x$与$\int_0^{\frac{\pi}{2}}\cos^9x\textrm{d}x$。\medskip 原式1为偶数次幂,所以$=\dfrac{9}{10}\cdot\dfrac{7}{8}\cdot\dfrac{5}{6}\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{\pi}{2}\cdot\dfrac{9!!}{10!!}$。\medskip @@ -1027,7 +1027,7 @@ $a,b,c>0$: \item $\sqrt[3]{abc}\leqslant\dfrac{a+b+c}{3}\leqslant\dfrac{a^2+b^2+c^2}{3}$。 \end{enumerate} -\textbf{例题6:}证明函数$f(x)=\dfrac{x}{1+x^2}$在$(-\infty,+\infty)$内有界。 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0}\frac{1}{x^2}\ln\frac{\tan x}{x}}$ - -$=e^{\lim\limits_{x\to 0}\frac{1}{x^2}\frac{\tan x-x}{x}}$ - -$=e^{\lim\limits_{x\to 0}\frac{1}{x^2}\cdot\frac{x^2}{3}}$ - -$= e^{\frac{1}{3}}$ +$\therefore e^{\lim\limits_{x\to 0}\frac{1}{x^2}\ln\frac{\tan x}{x}}=e^{\lim\limits_{x\to 0}\frac{1}{x^2}\frac{\tan x-x}{x}}=e^{\lim\limits_{x\to 0}\frac{1}{x^2}\cdot\frac{x^2}{3}}=e^{\frac{1}{3}}$ 根据海涅定理:取$x=\dfrac{1}{n},n\to\infty$,$\lim\limits_{n\to\infty}\left(n\tan\dfrac{1}{n}\right)^{n^2}=e^{\frac{1}{3}}$。 @@ -887,21 +879,9 @@ $\therefore\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$。 证明: -$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x$\medskip +$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=\lim\limits_{x\to\infty}e^{\ln(1+\frac{1}{x})^x}=\lim\limits_{x\to\infty}e^{x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}\frac{\ln(1+\frac{1}{x})}{\frac{1}{x}}}$ -$=\lim\limits_{x\to\infty}e^{\ln(1+\frac{1}{x})^x}$ - -$=\lim\limits_{x\to\infty}e^{x\ln(1+\frac{1}{x})}$ - -$=e^{\lim\limits_{x\to\infty}x\ln(1+\frac{1}{x})}$ - -$=e^{\lim\limits_{x\to\infty}\frac{\ln(1+\frac{1}{x})}{\frac{1}{x}}}$ - -$=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}}}$ - -$=e^{\lim\limits_{x\to\infty}\frac{1}{1+x}}$ - -$=e$ +$=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}}}=e^{\lim\limits_{x\to\infty}\frac{1}{1+x}}=e$\medskip 从而$\lim\limits_{\Delta\to\infty}\left(1+\dfrac{1}{\Delta}\right)^\Delta=e$与$\lim\limits_{\Delta\to 0}\left(1+\Delta\right)^{\frac{1}{\Delta}}=e(\Delta\neq 0)$。 diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf index 66d875de626a03c0bb97130ecee93ff2ad7e070f..50b5f8e4f4a1862000867bbbab06b498b48aad01 100644 GIT binary patch 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b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex index 1866873..66d75b5 100644 --- a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -446,19 +446,13 @@ $\textrm{d}S=\left[(y+4)-\dfrac{y^2}{2}\right]\textrm{d}y$。 因为无法计算对于$x$的表达式,所以使用参数方程代入,并改变上下限$S$: -$=\int_0^{2\pi}a(1-\cos t)\,\textrm{d}[a(t-\sin t)]$ +$=\int_0^{2\pi}a(1-\cos t)\,\textrm{d}[a(t-\sin t)]=\int_0^{2\pi}a^2(1-\cos t)^2\,\textrm{d}t$ -$=\int_0^{2\pi}a^2(1-\cos t)^2\,\textrm{d}t$ - -$=a^2\displaystyle{\int_0^{2\pi}\left(2\sin^2\dfrac{t}{2}\right)^2\textrm{d}t}$(消去里面的1) - -$=4a^2\displaystyle{\int_0^{2\pi}\sin^4\dfrac{t}{2}\,\textrm{d}t}$ +$=a^2\displaystyle{\int_0^{2\pi}\left(2\sin^2\dfrac{t}{2}\right)^2\textrm{d}t}\,\text{(消去里面的1)}=4a^2\displaystyle{\int_0^{2\pi}\sin^4\dfrac{t}{2}\,\textrm{d}t}$ 令$u=\dfrac{t}{2}$,从而$\textrm{d}t=2\textrm{d}u$,从而$u\in[0,\pi]$。 -$=8a^2\int_0^\pi\sin^4u\,\textrm{d}u$ - -$=16a^2\int_0^{\frac{\pi}{2}}\sin^4u\,\textrm{d}u$(积分可加性拆分为两个相同限的项) +$=8a^2\int_0^\pi\sin^4u\,\textrm{d}u=16a^2\int_0^{\frac{\pi}{2}}\sin^4u\,\textrm{d}u$(积分可加性拆分为两个相同限的项) $=16a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=3a^2\pi$(点火公式)。 @@ -476,13 +470,9 @@ $=16a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=3a^2\pi$(点火 所以可以根据公式$S=2\dfrac{1}{2}\int_0^\pi a^2(1+\cos\theta)^2\,\textrm{d}\theta$。 -$=a^2\displaystyle{\int_0^\pi\left(2\cos^2\dfrac{\theta}{2}\right)^2\textrm{d}\theta}$ +$=a^2\displaystyle{\int_0^\pi\left(2\cos^2\dfrac{\theta}{2}\right)^2\textrm{d}\theta}=4a^2\displaystyle{\int_0^\pi\cos^4\dfrac{\theta}{2}\,\textrm{d}\theta}$ -$=4a^2\displaystyle{\int_0^\pi\cos^4\dfrac{\theta}{2}\,\textrm{d}\theta}$ - -令$\dfrac{\theta}{2}=t$,所以$\textrm{d}\theta=2\textrm{d}t$,同时上下限缩小一半: - -$=8a^2\int_0^{\frac{\pi}{2}}\cos^4t\,\textrm{d}t$ +令$\dfrac{\theta}{2}=t$,所以$\textrm{d}\theta=2\textrm{d}t$,同时上下限缩小一半:$=8a^2\int_0^{\frac{\pi}{2}}\cos^4t\,\textrm{d}t$ 根据华理士公式:$=8a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{3}{2}a^2\pi$。 @@ -514,27 +504,19 @@ $V_x=\pi\int_0^{2a\pi}y^2\,\textrm{d}x$ 代入参数方程并改变上下限: -$=\pi\int_0^{2\pi}a^2(1-\cos t)^2\,\textrm{d}[a(t-\sin t)]$ +$=\pi\int_0^{2\pi}a^2(1-\cos t)^2\,\textrm{d}[a(t-\sin t)]=a^3\pi\int_0^{2\pi}(1-\cos t)^3\,\textrm{d}t$ -$=a^3\pi\int_0^{2\pi}(1-\cos t)^3\,\textrm{d}t$ - -$=a^3\pi\displaystyle{\int_0^{2\pi}\left(2\sin^2\dfrac{t}{2}\right)^3\textrm{d}t}$ - -$=8a^3\pi\displaystyle{\int_0^{2\pi}\sin^6\dfrac{t}{2}\textrm{d}t}$ +$=a^3\pi\displaystyle{\int_0^{2\pi}\left(2\sin^2\dfrac{t}{2}\right)^3\textrm{d}t}=8a^3\pi\displaystyle{\int_0^{2\pi}\sin^6\dfrac{t}{2}\textrm{d}t}$ 令$\dfrac{\theta}{2}=t$,所以$\textrm{d}\theta=2\textrm{d}t$,同时上下限缩小一半: -$=16a^3\pi\int_0^\pi\sin^6u\,\textrm{d}u$ - -$=32a^3\pi\int_0^{\frac{\pi}{2}}\sin^6u\,\textrm{d}u$ +$=16a^3\pi\int_0^\pi\sin^6u\,\textrm{d}u=32a^3\pi\int_0^{\frac{\pi}{2}}\sin^6u\,\textrm{d}u$ 华理士公式得到最后$=5a^3\pi^2$。 同理可得$y$轴旋转体积为$V_y=2\pi\int_0^{2\pi}xy(x)\,\textrm{d}x$ -$=2\pi\int_0^{2\pi}a(t-\sin t)a^2(1-\cos t)^2\,\textrm{d}t$ - -$=2a^3\pi\int_0^{2\pi}(t-\sin t)\cdot 4\sin^4\dfrac{t}{2}\,\textrm{d}t$ +$=2\pi\int_0^{2\pi}a(t-\sin t)a^2(1-\cos t)^2\,\textrm{d}t=2a^3\pi\int_0^{2\pi}(t-\sin t)\cdot 4\sin^4\dfrac{t}{2}\,\textrm{d}t$ 然后拆开分别进行凑微分法,得到$6a^3\pi^3$。 @@ -564,4 +546,21 @@ $S_n=\sum\limits_{i=1}^n\Vert\overline{M_{i-1}M_{i}}\Vert$,$S=\lim\limits_{\de 如果是极坐标方程,则$S=\int_\alpha^\beta\sqrt{\rho^2+\rho'^2}\,\textrm{d}\theta$。 +\section{积分表} + +\begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + 原函数 & 积分函数 & 原函数 & 积分函数\\ \hline + $\int k\,\textrm{d}x$ & $kx+C$ & $\int x^\mu\,\textrm{d}x$ & $\dfrac{x^{\mu+1}}{\mu+1}+C$ \\ \hline + $\int\dfrac{\textrm{d}x}{x}$ & $\ln\vert x\vert+C$ & $\int\dfrac{\textrm{d}x}{1+x^2}$ & $\arctan x+C$ \\ \hline + $\int\dfrac{\textrm{d}x}{\sqrt{1-x^2}}$ & $\arcsin x+C$ & $\int\cos x\,\textrm{d}x$ & $\sin x+C$ \\ \hline + $\int\sin x\,\textrm{d}x$ & $-\cos x+C$ & $\int\dfrac{\textrm{d}x}{\cos^2x}$ & $\tan x+C$ \\ \hline + $\int\dfrac{\textrm{d}x}{\sin^2x}$ & $-\cot x+C$ & $\int\sec x\tan x\,\textrm{d}x$ & $\sec x+C$ \\ \hline + $\int\csc x\cot x\,\textrm{d}x$ & $-\csc x+C$ & $\int e^x\,\textrm{d}x$ & $e^x+C$ \\ \hline + $\int a^x\,\textrm{d}x$ & $\dfrac{a^x}{\ln a}+C$ & & \\ + \hline + \end{tabular} +\end{center} + \end{document} diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf 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f'(\dfrac{3}{2})>f''(0)>f(-\dfrac{1}{2})$。\medskip \textbf{例题:}$\left(x^\alpha\right)'=\alpha x^{\alpha-1}(x>0)$。\medskip -$\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$\medskip +$\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{\left(x+\Delta x\right)^\alpha-x^\alpha}{\Delta x}$\medskip -$=\lim\limits_{\Delta x\to 0}\dfrac{\left(x+\Delta x\right)^\alpha-x^\alpha}{\Delta x}$\medskip - -$=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\left[\left(1+\dfrac{\Delta x}{x}\right)^\alpha-1\right]}{\Delta x}$\medskip - -$=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\cdot\alpha\cdot\dfrac{\Delta x}{x}}{\Delta x}$\medskip - -$=\alpha x^{\alpha-1}$ +$=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\left[\left(1+\dfrac{\Delta x}{x}\right)^\alpha-1\right]}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\cdot\alpha\cdot\dfrac{\Delta x}{x}}{\Delta x}=\alpha x^{\alpha-1}$ \subsection{导数的几何意义} @@ -144,13 +138,7 @@ $=\alpha x^{\alpha-1}$ 可导定义:$f'(x)=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x} = A$ -$\lim\limits_{\Delta x\to 0}f(x+\Delta x)-f(x)$ - -$=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\cdot\Delta x$ - -$=A\cdot 0$ - -$=0$ +$\lim\limits_{\Delta x\to 0}f(x+\Delta x)-f(x)=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\cdot\Delta x=A\cdot 0=0$ \textbf{例题:}若$f(x)$在$x=x_0$处连续,且$\lim\limits_{x\to x_0}\dfrac{f(x)}{x-x_0}=A$,则$f(x_0)=0$且$f'(x_0)=A$。 @@ -178,11 +166,7 @@ $=0$ $(u\cdot v)'$ -$=f'(x)$ - -$=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$ - -$=\lim\limits_{\Delta x\to 0}\dfrac{u(x+\Delta x)v(x+\Delta x)-u(x)v(x)}{\Delta x}$ +$=f'(x)=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{u(x+\Delta x)v(x+\Delta x)-u(x)v(x)}{\Delta x}$ $=\lim\limits_{\Delta x\to 0}\dfrac{u(x+\Delta x)v(x+\Delta x)-u(x)v(x+\Delta x)+u(x)v(x+\Delta x)-u(x)v(x)}{\Delta x}$ @@ -210,23 +194,11 @@ $\therefore\dfrac{\textrm{d}\arcsin x}{\textrm{d}x}=\dfrac{1}{\dfrac{\textrm{d}\ $\therefore\dfrac{\textrm{d}\arctan x}{\textrm{d}x}=\dfrac{1}{\dfrac{\textrm{d}\tan y}{\textrm{d}y}}=\dfrac{1}{\sec^2y}=\dfrac{1}{1+\tan^2y}=\dfrac{1}{1+x^2}$。\medskip -二阶反函数导数\textcolor{aqua}{\textbf{定理:}}: +二阶反函数导数\textcolor{aqua}{\textbf{定理:}}:\medskip -$f''(x)$ +$f''(x)=y''_{xx}=\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y}{\textrm{d}x}\right)}{\textrm{d}x}=\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}\left(\dfrac{1}{\varphi'(y)}\right)}{\textrm{d}x}=\dfrac{\textrm{d}\left(\dfrac{1}{\varphi'(y)}\right)}{\textrm{d}y}\cdot\dfrac{\textrm{d}y}{\textrm{d}x}$ -$=y''_{xx}$\medskip - -$=\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y}{\textrm{d}x}\right)}{\textrm{d}x}$\medskip - -$=\dfrac{\textrm{d}^2y}{\textrm{d}x^2}$\medskip - -$=\dfrac{\textrm{d}\left(\dfrac{1}{\varphi'(y)}\right)}{\textrm{d}x}$\medskip - -$=\dfrac{\textrm{d}\left(\dfrac{1}{\varphi'(y)}\right)}{\textrm{d}y}\cdot\dfrac{\textrm{d}y}{\textrm{d}x}$\medskip - -$=-\dfrac{x_{yy}''}{(x_y')^2}\cdot\dfrac{1}{x_y'}$\medskip - -$=-\dfrac{x_{yy}''}{(x_y')^3}$\medskip +$=-\dfrac{x_{yy}''}{(x_y')^2}\cdot\dfrac{1}{x_y'}=-\dfrac{x_{yy}''}{(x_y')^3}$\medskip 其中$\textrm{d}x\cdot\textrm{d}x=(\textrm{d}x)^2=\textrm{d}x^2$称为微分的幂,而$\textrm{d}(x^2)$叫幂的微分。 @@ -256,15 +228,7 @@ $\therefore$根据导数的四则运算,需要导数的乘积为每一项求 $\therefore f'(1)$ -$=f'(x)\vert_{x=1}$\medskip - -$=\dfrac{\pi}{2}\cdot g(1)+0\cdot g'(x)$\medskip - -$=\dfrac{\pi}{2}\cdot g(1)$\medskip - -$=\dfrac{\pi}{2}(-1)(-2)\cdots(-99)$ - -$=-\dfrac{\pi}{2}\cdot 99!$ +$=f'(x)\vert_{x=1}=\dfrac{\pi}{2}\cdot g(1)+0\cdot g'(x)=\dfrac{\pi}{2}\cdot g(1)=\dfrac{\pi}{2}(-1)(-2)\cdots(-99)=-\dfrac{\pi}{2}\cdot 99!$ \subsection{分段函数的导数} diff --git 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