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如何设计网站建设引导页,系统平台,做网站 徐州,在北京网站建设的岗位微积分期末复习提纲详解 一、极限#xff08;Limit Review#xff09; 1. 定义 设函数 f(x)f(x)f(x) 在点 x0x_0x0​ 的某个去心邻域内有定义#xff0c;如果存在常数 AAA#xff0c;对于任意给定的正数 ε\varepsilonε#xff0c;总存在正数 δ\deltaδ#xff0c;使得…微积分期末复习提纲详解一、极限Limit Review1. 定义设函数f(x)f(x)f(x)在点x0x_0x0​的某个去心邻域内有定义如果存在常数AAA对于任意给定的正数ε\varepsilonε总存在正数δ\deltaδ使得当0∣x−x0∣δ0 |x - x_0| \delta0∣x−x0​∣δ时有∣f(x)−A∣ε|f(x) - A| \varepsilon∣f(x)−A∣ε则称AAA是函数f(x)f(x)f(x)当x→x0x \to x_0x→x0​时的极限记作lim⁡x→x0f(x)A \lim_{x \to x_0} f(x) Ax→x0​lim​f(x)A2. 极限的性质唯一性若极限存在则极限值唯一。局部有界性若极限存在则函数在某个去心邻域内有界。局部保号性若lim⁡x→x0f(x)A0\lim_{x \to x_0} f(x) A 0limx→x0​​f(x)A0则存在某个去心邻域使得在该邻域内f(x)0f(x) 0f(x)0。四则运算法则设lim⁡f(x)A\lim f(x) Alimf(x)Alim⁡g(x)B\lim g(x) Blimg(x)B则lim⁡[f(x)±g(x)]A±B,lim⁡[f(x)⋅g(x)]A⋅B,lim⁡f(x)g(x)AB (B≠0) \lim [f(x) \pm g(x)] A \pm B, \quad \lim [f(x) \cdot g(x)] A \cdot B, \quad \lim \frac{f(x)}{g(x)} \frac{A}{B} \ (B \neq 0)lim[f(x)±g(x)]A±B,lim[f(x)⋅g(x)]A⋅B,limg(x)f(x)​BA​(B0)3. 求解技巧1夹逼定理如果g(x)≤f(x)≤h(x)g(x) \le f(x) \le h(x)g(x)≤f(x)≤h(x)且lim⁡g(x)lim⁡h(x)A\lim g(x) \lim h(x) Alimg(x)limh(x)A则lim⁡f(x)A\lim f(x) Alimf(x)A。2有理化分子常用于含有根式的极限例如lim⁡x→01x−1xlim⁡x→0(1x−1)(1x1)x(1x1)lim⁡x→01212 \lim_{x \to 0} \frac{\sqrt{1x} - 1}{x} \lim_{x \to 0} \frac{(\sqrt{1x} - 1)(\sqrt{1x} 1)}{x(\sqrt{1x} 1)} \lim_{x \to 0} \frac{1}{2} \frac{1}{2}x→0lim​x1x​−1​x→0lim​x(1x​1)(1x​−1)(1x​1)​x→0lim​21​21​3洛必达法则适用于00\frac{0}{0}00​或∞∞\frac{\infty}{\infty}∞∞​型极限。若lim⁡x→af(x)g(x)\lim_{x \to a} \frac{f(x)}{g(x)}limx→a​g(x)f(x)​满足条件则lim⁡x→af(x)g(x)lim⁡x→af′(x)g′(x) \lim_{x \to a} \frac{f(x)}{g(x)} \lim_{x \to a} \frac{f(x)}{g(x)}x→alim​g(x)f(x)​x→alim​g′(x)f′(x)​4等价无穷小当x→0x \to 0x→0时常用等价无穷小替换注意只能用于乘积因子sin⁡x∼x\sin x \sim xsinx∼xtan⁡x∼x\tan x \sim xtanx∼xarcsin⁡x∼x\arcsin x \sim xarcsinx∼xln⁡(1x)∼x\ln(1x) \sim xln(1x)∼xex−1∼xe^x - 1 \sim xex−1∼x1−cos⁡x∼12x21 - \cos x \sim \frac{1}{2}x^21−cosx∼21​x25指数型极限的对数求法对于形如lim⁡f(x)g(x)\lim f(x)^{g(x)}limf(x)g(x)的未定式如1∞1^\infty1∞、000^000、∞0\infty^0∞0设yf(x)g(x)y f(x)^{g(x)}yf(x)g(x)取对数ln⁡yg(x)ln⁡f(x) \ln y g(x) \ln f(x)lnyg(x)lnf(x)先求lim⁡ln⁡y\lim \ln ylimlny则原极限为elim⁡ln⁡ye^{\lim \ln y}elimlny。6大O与小o记号f(x)O(g(x))f(x) O(g(x))f(x)O(g(x))表示存在常数CCC使得∣f(x)∣≤C∣g(x)∣|f(x)| \le C|g(x)|∣f(x)∣≤C∣g(x)∣在某个邻域内成立。f(x)o(g(x))f(x) o(g(x))f(x)o(g(x))表示lim⁡f(x)g(x)0\lim \frac{f(x)}{g(x)} 0limg(x)f(x)​0。4. 例题计算lim⁡x→0x−sin⁡xx3⋅xarcsin⁡x⋅xex−1 \lim_{x \to 0} \frac{x - \sin x}{x^3} \cdot \frac{x}{\arcsin x} \cdot \frac{x}{e^x - 1}x→0lim​x3x−sinx​⋅arcsinxx​⋅ex−1x​解利用等价无穷小当x→0x \to 0x→0时arcsin⁡x∼x\arcsin x \sim xarcsinx∼xex−1∼xe^x - 1 \sim xex−1∼x所以后两项极限均为1。对第一项使用洛必达法则或泰勒展开lim⁡x→0x−sin⁡xx3lim⁡x→01−cos⁡x3x2lim⁡x→0sin⁡x6x16 \lim_{x \to 0} \frac{x - \sin x}{x^3} \lim_{x \to 0} \frac{1 - \cos x}{3x^2} \lim_{x \to 0} \frac{\sin x}{6x} \frac{1}{6}x→0lim​x3x−sinx​x→0lim​3x21−cosx​x→0lim​6xsinx​61​因此原极限为16\frac{1}{6}61​。二、连续性Continuity1. 定义设函数f(x)f(x)f(x)在点x0x_0x0​的某邻域内有定义若lim⁡x→x0f(x)f(x0)\lim_{x \to x_0} f(x) f(x_0)limx→x0​​f(x)f(x0​)则称f(x)f(x)f(x)在x0x_0x0​连续。2. 复合函数连续性若fff在x0x_0x0​连续ggg在f(x0)f(x_0)f(x0​)连续则复合函数g∘fg \circ fg∘f在x0x_0x0​连续。3. 介值定理IVT若fff在闭区间[a,b][a,b][a,b]上连续且f(a)≠f(b)f(a) \ne f(b)f(a)f(b)则对于任意介于f(a)f(a)f(a)和f(b)f(b)f(b)之间的数CCC存在至少一点ξ∈(a,b)\xi \in (a,b)ξ∈(a,b)使得f(ξ)Cf(\xi) Cf(ξ)C。4. 极值定理EVT若fff在闭区间[a,b][a,b][a,b]上连续则fff在[a,b][a,b][a,b]上必取得最大值和最小值。5. 间断点类型第一类间断点左右极限都存在。若相等但不等于函数值或函数值无定义则为可去间断点若不相等则为跳跃间断点。第二类间断点左右极限至少有一个不存在如无穷间断点、振荡间断点。6. 可积性若fff在[a,b][a,b][a,b]上连续则fff在[a,b][a,b][a,b]上可积。有界且只有有限个间断点的函数也可积。三、导数Derivatives1. 定义函数yf(x)y f(x)yf(x)在点x0x_0x0​处的导数定义为f′(x0)lim⁡Δx→0f(x0Δx)−f(x0)Δxlim⁡x→x0f(x)−f(x0)x−x0 f(x_0) \lim_{\Delta x \to 0} \frac{f(x_0\Delta x) - f(x_0)}{\Delta x} \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}f′(x0​)Δx→0lim​Δxf(x0​Δx)−f(x0​)​x→x0​lim​x−x0​f(x)−f(x0​)​若该极限存在则称函数在x0x_0x0​可导。2. 线性化在点x0x_0x0​附近用切线近似函数值L(x)f(x0)f′(x0)(x−x0) L(x) f(x_0) f(x_0)(x - x_0)L(x)f(x0​)f′(x0​)(x−x0​)3. 中值定理MVT若fff在[a,b][a,b][a,b]上连续在(a,b)(a,b)(a,b)内可导则存在至少一点ξ∈(a,b)\xi \in (a,b)ξ∈(a,b)使得f′(ξ)f(b)−f(a)b−a f(\xi) \frac{f(b) - f(a)}{b - a}f′(ξ)b−af(b)−f(a)​4. 函数作图利用导数分析函数性质一阶导数f′(x)f(x)f′(x)确定单调区间f′(x)0f(x) 0f′(x)0递增f′(x)0f(x) 0f′(x)0递减和极值点临界点处f′(x)0f(x) 0f′(x)0或不存在。二阶导数f′′(x)f(x)f′′(x)确定凹凸性f′′(x)0f(x) 0f′′(x)0凹向上f′′(x)0f(x) 0f′′(x)0凹向下和拐点。5. 求导法则四则运算(u±v)′u′±v′,(uv)′u′vuv′,(uv)′u′v−uv′v2 (u \pm v) u \pm v, \quad (uv) uv uv, \quad \left(\frac{u}{v}\right) \frac{uv - uv}{v^2}(u±v)′u′±v′,(uv)′u′vuv′,(vu​)′v2u′v−uv′​链式法则ddxf(g(x))f′(g(x))⋅g′(x) \frac{d}{dx} f(g(x)) f(g(x)) \cdot g(x)dxd​f(g(x))f′(g(x))⋅g′(x)6. 牛顿法用于求方程f(x)0f(x) 0f(x)0的近似解迭代公式xn1xn−f(xn)f′(xn) x_{n1} x_n - \frac{f(x_n)}{f(x_n)}xn1​xn​−f′(xn​)f(xn​)​7. 反函数求导若yf(x)y f(x)yf(x)可逆且f′(x)≠0f(x) \ne 0f′(x)0则反函数xf−1(y)x f^{-1}(y)xf−1(y)的导数为dxdy1f′(x)1f′(f−1(y)) \frac{dx}{dy} \frac{1}{f(x)} \frac{1}{f(f^{-1}(y))}dydx​f′(x)1​f′(f−1(y))1​8. 隐函数求导对由方程F(x,y)0F(x,y) 0F(x,y)0确定的隐函数两边对xxx求导然后解出dydx\frac{dy}{dx}dxdy​。例如对x2y21x^2 y^2 1x2y21求导得2x2ydydx02x 2y \frac{dy}{dx} 02x2ydxdy​0所以dydx−xy\frac{dy}{dx} -\frac{x}{y}dxdy​−yx​。9. 对数求导法对形如yf(x)g(x)y f(x)^{g(x)}yf(x)g(x)或连乘除的函数先取对数ln⁡yln⁡f(x)\ln y \ln f(x)lnylnf(x)或ln⁡yg(x)ln⁡f(x)\ln y g(x) \ln f(x)lnyg(x)lnf(x)然后两边对xxx求导解出y′yy′。四、原函数与不定积分Anti-derivatives1. 三角函数积分基本积分公式∫sin⁡x dx−cos⁡xC\int \sin x \, dx -\cos x C∫sinxdx−cosxC∫cos⁡x dxsin⁡xC\int \cos x \, dx \sin x C∫cosxdxsinxC∫sec⁡2x dxtan⁡xC\int \sec^2 x \, dx \tan x C∫sec2xdxtanxC∫csc⁡2x dx−cot⁡xC\int \csc^2 x \, dx -\cot x C∫csc2xdx−cotxC∫sec⁡xtan⁡x dxsec⁡xC\int \sec x \tan x \, dx \sec x C∫secxtanxdxsecxC∫csc⁡xcot⁡x dx−csc⁡xC\int \csc x \cot x \, dx -\csc x C∫cscxcotxdx−cscxC2. 换元法第一类换元凑微分∫f(φ(x))φ′(x) dx∫f(u) du\int f(\varphi(x)) \varphi(x) \, dx \int f(u) \, du∫f(φ(x))φ′(x)dx∫f(u)du其中uφ(x)u \varphi(x)uφ(x)。第二类换元常用于根式积分例如a2−x2\sqrt{a^2 - x^2}a2−x2​令xasin⁡tx a \sin txasinta2x2\sqrt{a^2 x^2}a2x2​令xatan⁡tx a \tan txatantx2−a2\sqrt{x^2 - a^2}x2−a2​令xasec⁡tx a \sec txasect3. 有理函数积分将有理函数分解为部分分式之和然后逐项积分。例如∫P(x)Q(x) dx \int \frac{P(x)}{Q(x)} \, dx∫Q(x)P(x)​dx其中P(x)P(x)P(x)、Q(x)Q(x)Q(x)为多项式若deg⁡Pdeg⁡Q\deg P \deg QdegPdegQ可进行部分分式分解。4. 分部积分法公式∫u dvuv−∫v du \int u \, dv uv - \int v \, du∫udvuv−∫vdu选择uuu的顺序可参考“反对幂指三”反三角函数、对数函数、幂函数、指数函数、三角函数通常将优先级高的选为uuu。五、定积分Definite Integral1. 定义黎曼和设函数f(x)f(x)f(x)在区间[a,b][a,b][a,b]上有定义将区间任意分割取样本点和式极限∫abf(x) dxlim⁡max⁡Δxi→0∑i1nf(ξi)Δxi \int_a^b f(x) \, dx \lim_{\max \Delta x_i \to 0} \sum_{i1}^n f(\xi_i) \Delta x_i∫ab​f(x)dxmaxΔxi​→0lim​i1∑n​f(ξi​)Δxi​2. 微积分基本定理FTC第一部分若fff在[a,b][a,b][a,b]上连续则函数F(x)∫axf(t) dtF(x) \int_a^x f(t) \, dtF(x)∫ax​f(t)dt在[a,b][a,b][a,b]上可导且F′(x)f(x)F(x) f(x)F′(x)f(x)。第二部分若FFF是fff的一个原函数则∫abf(x) dxF(b)−F(a) \int_a^b f(x) \, dx F(b) - F(a)∫ab​f(x)dxF(b)−F(a)3. 换元法与不定积分类似但需注意换限。设xφ(t)x \varphi(t)xφ(t)则∫abf(x) dx∫φ−1(a)φ−1(b)f(φ(t))φ′(t) dt \int_a^b f(x) \, dx \int_{\varphi^{-1}(a)}^{\varphi^{-1}(b)} f(\varphi(t)) \varphi(t) \, dt∫ab​f(x)dx∫φ−1(a)φ−1(b)​f(φ(t))φ′(t)dt4. 分部积分法∫abu dv[uv]ab−∫abv du \int_a^b u \, dv [uv]_a^b - \int_a^b v \, du∫ab​udv[uv]ab​−∫ab​vdu5. 面积与体积平面图形面积由曲线yf(x)y f(x)yf(x)、yg(x)y g(x)yg(x)和直线xax axa、xbx bxb围成面积为A∫ab∣f(x)−g(x)∣ dx A \int_a^b |f(x) - g(x)| \, dxA∫ab​∣f(x)−g(x)∣dx旋转体体积绕 x 轴旋转Vπ∫ab[f(x)]2 dxV \pi \int_a^b [f(x)]^2 \, dxVπ∫ab​[f(x)]2dx绕 y 轴旋转柱壳法V2π∫abxf(x) dxV 2\pi \int_a^b x f(x) \, dxV2π∫ab​xf(x)dx6. 弧长曲线yf(x)y f(x)yf(x)a≤x≤ba \le x \le ba≤x≤b的弧长为L∫ab1[f′(x)]2 dx L \int_a^b \sqrt{1 [f(x)]^2} \, dxL∫ab​1[f′(x)]2​dx7. 数值积分梯形法∫abf(x) dx≈b−a2n[f(x0)2∑i1n−1f(xi)f(xn)]\int_a^b f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) 2\sum_{i1}^{n-1} f(x_i) f(x_n) \right]∫ab​f(x)dx≈2nb−a​[f(x0​)2∑i1n−1​f(xi​)f(xn​)]辛普森法∫abf(x) dx≈b−a6n[f(x0)4∑i1nf(x2i−1)2∑i1n−1f(x2i)f(x2n)]\int_a^b f(x) \, dx \approx \frac{b-a}{6n} \left[ f(x_0) 4\sum_{i1}^{n} f(x_{2i-1}) 2\sum_{i1}^{n-1} f(x_{2i}) f(x_{2n}) \right]∫ab​f(x)dx≈6nb−a​[f(x0​)4∑i1n​f(x2i−1​)2∑i1n−1​f(x2i​)f(x2n​)]8. 反常积分1无穷区间∫a∞f(x) dxlim⁡t→∞∫atf(x) dx \int_a^{\infty} f(x) \, dx \lim_{t \to \infty} \int_a^t f(x) \, dx∫a∞​f(x)dxt→∞lim​∫at​f(x)dx2无界函数若fff在aaa点无界则∫abf(x) dxlim⁡t→a∫tbf(x) dx \int_a^b f(x) \, dx \lim_{t \to a^} \int_t^b f(x) \, dx∫ab​f(x)dxt→alim​∫tb​f(x)dx3p-判别法∫1∞1xp dx\int_1^{\infty} \frac{1}{x^p} \, dx∫1∞​xp1​dx收敛当且仅当p1p 1p1∫011xp dx\int_0^1 \frac{1}{x^p} \, dx∫01​xp1​dx收敛当且仅当p1p 1p14比较判别法若0≤f(x)≤g(x)0 \le f(x) \le g(x)0≤f(x)≤g(x)在[a,∞)[a, \infty)[a,∞)上则若∫a∞g(x) dx\int_a^{\infty} g(x) \, dx∫a∞​g(x)dx收敛则∫a∞f(x) dx\int_a^{\infty} f(x) \, dx∫a∞​f(x)dx收敛若∫a∞f(x) dx\int_a^{\infty} f(x) \, dx∫a∞​f(x)dx发散则∫a∞g(x) dx\int_a^{\infty} g(x) \, dx∫a∞​g(x)dx发散六、常微分方程ODE1. 斜率场给定微分方程dydxf(x,y)\frac{dy}{dx} f(x,y)dxdy​f(x,y)在平面区域上每一点(x,y)(x,y)(x,y)画出斜率为f(x,y)f(x,y)f(x,y)的短线段形成斜率场。解曲线积分曲线在每点处与斜率场的线段相切。2. 可分离变量方程形如dydxg(x)h(y)\frac{dy}{dx} g(x)h(y)dxdy​g(x)h(y)分离变量dyh(y)g(x) dx \frac{dy}{h(y)} g(x) \, dxh(y)dy​g(x)dx两边积分求解。3. 线性微分方程一阶线性方程dydxP(x)yQ(x)\frac{dy}{dx} P(x)y Q(x)dxdy​P(x)yQ(x)通解公式ye−∫P(x) dx(∫Q(x)e∫P(x) dx dxC) y e^{-\int P(x) \, dx} \left( \int Q(x) e^{\int P(x) \, dx} \, dx C \right)ye−∫P(x)dx(∫Q(x)e∫P(x)dxdxC)4. 自治方程与相线分析自治方程dydtf(y)\frac{dy}{dt} f(y)dtdy​f(y)不显含自变量ttt。相线分析步骤求平衡点f(y)0f(y) 0f(y)0的根判断稳定性若f′(y0)0f(y_0) 0f′(y0​)0则平衡点稳定若f′(y0)0f(y_0) 0f′(y0​)0则不稳定5. 数学模型指数增长/衰减dydtky\frac{dy}{dt} kydtdy​ky解为yy0ekty y_0 e^{kt}yy0​ektLogistic 模型dydtry(1−yK)\frac{dy}{dt} ry \left(1 - \frac{y}{K}\right)dtdy​ry(1−Ky​)描述有限资源下的增长牛顿冷却定律dTdtk(T−Tenv)\frac{dT}{dt} k(T - T_{\text{env}})dtdT​k(T−Tenv​)物体温度随时间变化