托马斯微积分课件4.6 Substitution in Definite Integrals
微积分基本公式PPT学习教案
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例9. 设 f (x, y) xy et2 dt, 求 0
x y
2 f x2
2 2 f xy
y x
2 f y 2
.
解：令 (s) s et2 dt, 则 '(s) es2 , f (x, y) (xy) 0
于是
f (xy) '(xy) xy' ex2y2 y
e x x2 y2
2x3 yex2y2
最终结果 2ex2 y2
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例5 .
lim
x0
0 sin t 2dt
2x
x3
0 sin t 2dt '
lim 2x
x0 (x3 )'
lim
x0
s in(2 x ) 2 3x 2
(2 x )'
2 3
lim
x0
sin 4x x2
2
8 3
b( x)
f (t)dt
的导数 F( x) 为
a( x)
F( x) d
b( x)
f (t )dt
dx a( x)
f b( x)b( x)
f a( x)a( x)
证 F( x) 0 b( x) f (t)dt
a(x) 0
b( x) f (t )dt
a( x)
f (t)dt,
202
1
1 1 (2 2 1 1) 1
2
2
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例10 计算
2
f (x)dx, 其中
0
2x,
f
(
x)
5,
0 x1 1 x 2
解
2
托马斯微积分
Figure 2.43: The balloon in Example 3.
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 7
Figure 2.31: sin (x°) oscillates only /180 times as often as sin x oscillates. Its maximum slope is /180. (Example 9)
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 15
Figure 2.51: The position of the curve y = (a h – 1) /h, a > 0, varies continuously with a.
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 16
Functionals and stocpresentation of martingales Rama Cont泛函Ito微积分与鞅的随机积分表示(英文版)
Keywords: stochastic calculus, functional calculus, Ito formula, integration by parts, Malliavin derivative, martingale representation, semimartingale, Wiener functionals, functional Feynman-Kac formula, Kolmogorov equation, Clark-Ocone formula.
2
1
Introduction
Ito’s stochastic calculus [15, 16, 8, 24, 20, 28] has proven to be a powerful and useful tool in analyzing phenomena involving random, irregular evolution in time. Two characteristics distinguish the Ito calculus from other approaches to integration, which may also apply to stochastic processes. First is the possibility of dealing with processes, such as Brownian motion, which have non-smooth trajectories with infinite variation. Second is the non-anticipative nature of the quantities involved: viewed as a functional on the space of paths indexed by time, a non-anticipative quantity may only depend on the underlying path up to the current time. This notion, first formalized by Doob [9] in the 1950s via the concept of a filtered probability space, is the mathematical counterpart to the idea of causality. Two pillars of stochastic calculus are the theory of stochastic integration, which allows to T define integrals 0 Y dX for of a large class of non-anticipative integrands Y with respect to a semimartingale X = (X (t), t ∈ [0, T ]), and the Ito formula [15, 16, 24] which allows to represent smooth functions Y (t) = f (t, X (t)) of a semimartingale in terms of such stochastic integrals. A central concept in both cases is the notion of quadratic variation [X ] of a semimartingale, which differentiates Ito calculus from the calculus of smooth functions. Whereas the class of integrands Y covers a wide range of non-anticipative path-dependent functionals of X , the Ito formula is limited to functions of the current value of X . Given that in many applications such as statistics of processes, physics or mathematical finance, one is led to consider functionals of a semimartingale X and its quadratic variation process [X ] such as:
微分方程差分方程(英文版)(托马斯微积分)
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嘉兴学院
6 March 2020
第十章 常微分方程与差分方程
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(3) y f ( y, y) 型
特点 不显含自变量 x. 解法 令 y P( x), y P dp ,
dy 代入原方程, 得 P dp f ( y, P).
dy
４.线性微分方程解的结构
（1） 二阶齐次方程解的结构:
形如 y P( x) y Q( x) y 0
(1)
嘉兴学院
6 March 2020
第十章 常微分方程与差分方程
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定理 1 如果函数 y1( x)与 y2 ( x)是方程(1)的两个
解,那末 y C1 y1 C2 y2 也是(1)的解.（C1, C2 是常 数）
定理 2：如果 y1( x)与 y2 ( x)是方程(1)的两个线性
嘉兴学院
6 March 2020
第十章 常微分方程与差分方程
第21页
定义2
含 有 未 知 函 数 两 个 或 两个 以 上 时 期 的 符 号 yx , yx1 , 的方程，称为差分方程.
形式：F ( x, yx , yx1, , yxn ) 0 或G( x, yx , yx1, , yxn ) 0 (n 1)
定理 3 设 yx* 是 n 阶常系数非齐次线性差分方程
yxn a1 yxn1 an1 yx1 an yx f x 2
的一个特解, Yx 是与(2)对应的齐次方程(1)的通
解,
那么
yx
Yx
y
* x
是
n
阶常系数非齐次线性差分
方程(2)的通解.
嘉兴学院
微积分CALCULUS.ppt
Solution:
a. Since g 32 ft/sec2 (9.8m/s2 ), V0 96 ft/sec and H0 112 ft, the height of the ball above the ground at time t is H (t) 16t 2 96t 112 feet. The velocity at time t is
c. Set v(t)=0, solve t=3. Thus, the ball is at its highest point when t=3 seconds.
d. The ball starts at H(0)=112 feet and rises to a maximum height of H(3)=256. So: The total distance travelled=2(256-112)+112 =400 feet.
acceleration acting on the object is the constant
downward acceleration g due to gravityistance is negligible). Thus, the height of the object
at time t is given by the formula
H
(t )
1 2
gt 2
V0t
H0
where H0 and V0 are the initial height and velocity of the object, respectively.
Example 10 Suppose a person standing at the top of a building 112 feet high throws a ball vertically upward with an initial velocity of 96 ft/sec.
微积分讲解ppt课件
多元函数的表示 方法
多元函数可用记号 f(x1,x2,…,xn)或z=f(x,y) 表示。
多元函数的定义 域
使多元函数有意义的自 变量组合(x1,x2,…,xn) 的集合。
多元函数的值域
多元函数所有值的集合 。
偏导数与全微分
偏导数的定义
设函数z=f(x,y)在点(x0,y0)的某一邻域内有定义，当y固定在y0而x在x0处有增量Δx时，相应地函数有增量 f(x0+Δx,y0)-f(x0,y0)。如果Δz与Δx之比当Δx→0时的极限存在，那么此极限值称为函数z=f(x,y)在点(x0,y0)处对 x的偏导数。
齐次方程法
通过变量替换，将齐次方程转化为可分离变 量的形式
一阶线性微分方程法
利用积分因子，将方程转化为可积分的形式
二阶常微分方程解法
可降阶的二阶微分方程
通过变量替换或分组，将方程降为一阶微分方 程求解
二阶线性微分方程法
利用特征根的性质，求解二阶线性常系数齐次 和非齐次微分方程
常系数线性微分方程组法
在经济学中的应用
边际分析
通过求导计算边际成本、边际收益等，为企业的决策 提供依据。
弹性分析
研究价格、需求等经济变量之间的相对变化关系，微 积分可用于计算弹性系数。
最优化问题
在资源有限的情况下，通过微积分求解最大化或最小 化某一经济指标的问题。
在工程学中的应用
结构力学
分析建筑、桥梁等结构的受力情况和稳定性，微积分可用 于求解复杂的力学方程。
通过消元法或特征根法，求解常系数线性微分方程组
05
多元函数微积分
多元函数的基本概念
多元函数的定义
设D为一个非空的n元有 序数组的集合，f为某一 确定的对应规则。若对 于每一个有序数组 (x1,x2,…,xn)∈D，通过 对应规则f，都有唯一确 定的实数y与之对应， 则称对应规则f为定义在 D上的n元函数。
微积分一些相关PPT
微分学
微分学主要研究的是在函数自变量变化时如
何确定函数值的瞬时变化率（或微分）。换 言之，计算导数的方法就叫微分学。微分学 的另一个计算方法是牛顿法，该算法又叫应 用几何法，主要通过函数曲线的切线来寻找 点斜率。费马常被称作“微分学的鼻祖”。
积分学
积分学是微分学的逆运算，即从导数推算出
原函数,又分为定积分与不定积分。一个一元 函数的定积分可以定义为无穷多小矩形的面 积和，约等于函数曲线下包含的实际面积。 因此，我们可以用积分来计算平面上一条曲 线所包含的面积、球体或圆锥体的表面积或 体积等。而不定积分的用途较少，主要用于 微分方程的解。
牛顿
牛顿在1671年写了《流数法和 无穷级数》，这本书直到1736 年才出版，它在这本书里指出： 变量是由点、线、面的连续运动产生的，否定了以 前自己认为的变量是无穷小元素的静止集合。他把 连续变量叫做流动量，把这些流动量的导数叫做流 数。牛顿在流数术中所提出的中心问题是：已知连 续运动的路径，求给定时刻的速度（微分法）；已 知运动的速度求给定时间内经过的路程（积分法）。
2：微积分的创立
微积分学的建立 从微积分成为一门学科来说，是在十七世纪，但是，微分和 积分的思想在古代就已经产生了。 极限的产生 公元前三世纪，古希腊的阿基米德在研究解决抛物弓形的面 积、球和球冠面积、螺线下面积和旋转双曲体的体积的问题 中，就隐含着近代积分学的思想。作为微分学基础的极限理 论来说，早在古代以有比较清楚的论述。比如中国的庄周所 著的《庄子》一书的“天下篇”中，记有“一尺之棰，日取 其半，万世不竭”。三国时期的刘徽在他的割圆术中提到 “割之弥细，所失弥小，割之又割，以至于不可割，则与圆 周和体而无所失矣。”这些都是朴素的、也是很典型的极限 概念。
托马斯微积分课件8.7 Taylor and Maclaurin Series
Analysis.
1 1 1 n 1 2 f x 2 x 2 3 x 2 n 1 x 2 2 2 2 2 12 1 x2 1 x 0, 4 1 x 2 2 x 2
n
1 n 1 2 n 1
1 1 2 x 1,1 , x , . 2 2
1 n 1 n n n 1 1 n f x x 2 x 1 2 x , x , . 3 n 0 2 2 n 0 3 n 0
目录
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Solution. The Taylor series generated by f at 0 is
The Taylor polynomial is
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Two questions: 1. How accurately do a function’s Taylor polynomial approximate the function on a given interval?
Step 3. Draw a conclusion.
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Example 5.
Solution.
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Example 6.
Solution.
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Example 7.
Solution.
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Example 10.
Solution.
x
托马斯微积分课件8.1 Limits os Sequences of Numbers
8.1 Limits of Sequences of Numbers 8.2 Subsequences, Bounded Sequences, and… 8.3 Infinite Series 8.4 Series of Nonnegative Terms 8.5 Alternating Series, Absolute and Conditional Convergence 8.6 Power Series 8.7 Taylor and Maclaurin Series 8.8 Applications of Power Series
n
பைடு நூலகம்
n
n 1 ln n 1
n
e
n ln
n 1 n 1
n 1 n e lim e n n 1 1 lim ln n 1 ln n 1 n e n
n 1 lim nln n n 1
ln n 4 n n 4 lim
1 n 4
e
1 lim ln n 4 n n 4
e
1 n n 4 lim
e
1
0
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Example 10. Applying the L’Hospital Rule
Solution.
n 1 e n 1
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Example 1. Applying the definition to show that
Solution. We must find the integer N such that for all n
微积分英文版课件
Applications of Derivatives
Local Extrema
Discover how derivatives help identify local maximums and minimums of functions.
Mean Value Theorem
Explore the mean value theorem and its applications in calculus.
Gradients and Directional Derivatives
2
derivatives and their applications in multivariable calculus.
Learn about gradients and
directional derivatives for
Derivatives
1
Definition of a Derivative
Uncover the definition and
Differentiability and Continuity
2
fundamental properties of derivatives.
Understand the relationship
Discover the conditions for a function to be continuous and its implications.
Explore the different types of discontinuities and their characteristics.
Conclusion
Review of Key Concepts
微积分讲解ppt课件
3.2.1 原函数和不定积分的概念
一、案例 二、概念和公式的引出
一、案例[路程函数]
已知物体的运动方程为 s(t) t2 ，则其速度为 v(t) s(t) (t 2 ) 2t
这里速度2t是路程t2的导数，反过来，路程t2又称为速 度2t的什么函数呢？若已知物体运动的速度v(t)，又如 何求物体的运动方程s(t)呢？
f xdx f x C 或 df x f x C
3.2.2 基本积分表
一、案例 二、概念和公式的引出
一、案例[幂函数的不定积分]
因为
x 1
1
x
x 1
1 是 x 的一个原函数
于是
x dx x 1 C
32微积分基本公式321原函数和不定积分的概念322基本积分表323微积分基本公式321原函数和不定积分的概念一案例二概念和公式的引出一案例路程函数已知物体的运动方程为又称为速度2t的什么函数呢
3.2 微积分基本公式
3.2.1 原函数和不定积分的概念 3.2.2 基本积分表 3.2.3 微积分基本公式
1
1
类似地, 由基本初等函数的求导公式，可以写出与之对应的不定积分公式．
二、概念和公式的引出
1.基本积分表
(1)
kdx kx C ( k 为常数）
(2) x dx x 1 C
1
1
(3)
1 x
dx
ln
x
C
(4) a xdx a x C
即两个函数和(差)的定积分等于它们定积分的和(差)． 性质１可推广到有限个函数的情形．
(2) 性质2 kf xdx k f xdx k为常数
高等数学（微积分学）专业术语名词概念定理等英汉对照
高等数学（微积分学）专业术语名词概念定理等英汉对照目录第一部分英汉微积分词汇Part 1 English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 function and Limi t (1)第二章导数与微分Chapter 2 Derivative and Differential (2)第三章微分中值定理Chapter 3 Mean Value theorem of differentials and the Application of Derivatives (3)第四章不定积分Chapter 4 Indefinite Intergrals (3)第五章定积分Chapter 5 Definite Integral (3)第六章定积分的应用Chapter 6 Application of the Definite Integrals (4)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geomertry and Vector Algebra (4) 第八章多元函数微分法及其应用Chapter 8 Differentiation of functions Several variables and Its Application (5)第九章重积分Multiple Integrals (6)第十章曲线积分与曲面积分Chapter 10 Line(Curve ) Integrals and Surface Integral s (6) 第十一章无穷级数Chapter 11 Infinite Series (6)第十二章微分方程Chapter 12 Differential Equation (7)第二部分定理定义公式的英文表达Part 2 English Expression for Theorem,Definition and Formula第一章函数与极限Chapter 1 Function and Limi t (19)1.1映射与函数(Mapping and Function ) (19)1.2数列的极限(Limit of the Sequence of Number) (20)1.3函数的极限(Limit of Function) (21)1.4无穷小与无穷大(Infinitesimal and Inifinity) (23)1.5极限运算法则(Operation Rule of Limit) (24)1.6极限存在准则两个重要的极限(Rule for theExistence of Limits Two Important Limits) (25)1.7无穷小的比较(The Comparison of infinitesimal) (26)1.8函数的连续性与间断点(Continuity of FunctionAnd Discontinuity Points) (28)1.9连续函数的运酸与初等函数的连续性(OperationOf Continuous Functions and Continuity ofElementary Functions) (28)1.10闭区间上联系汗水的性质(Properties ofContinuous Functions on a Closed Interval) (30)第二章导数与数分Chapter2 Derivative and Differential (31)2.1 导数的概念(The Concept of Derivative) (31)2.2 函数的求导法则(Rules for Finding Derivatives) (33)2.3 高阶导数(Higher-order Derivatives) (34)2.4 隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives ofImplicit Functions and Functions Determined by Parametric Equation andCorrelative Change Rate) (34)2.5 函数的微分(Differential of a Function) (35)第三章微分中值定理与导数的应用Chapter 3 Mean Value Theorem of Differentials and theApplication of Derivatives (36)3.1 微分中值定理(The Mean Value Theorem) (36)3.2 洛必达法则(L’Hopital’s Rule) (38)3.3 泰勒公式(Taylor’s Formula) (41)3.4 函数的单调性和曲线的凹凸性(Monotonicityof Functions and Concavity of Curves) (43)3.5 函数的极值与最大最小值(Extrema, Maximaand Minima of Functions) (46)3.6 函数图形的描绘(Graphing Functions) (49)3.7 曲率(Curvature) (50)3.8 方程的近似解(Solving Equation Numerically) (53)第四章不定积分Chapter 4Indefinite Integrals (54)4.1 不定积分的概念与性质(The Concept andProperties of Indefinite Integrals) (54)4.2 换元积分法(Substitution Rule for Indefinite Integrals) (56)4.3 分部积分法(Integration by Parts) (57)4.4 有理函数的积分(Integration of Rational Functions) (58)第五章定积分Chapter 5 Definite Integrals (61)5.1 定积分的概念和性质(Concept of Definite Integraland its Properties) (61)5.2 微积分基本定理(Fundamental Theorem of Calculus) (67)5.3 定积分的换元法和分部积分法(Integration by Substitution andDefinite Integrals by Parts) (69)5.4 反常积分(Improper Integrals) (70)第六章定积分的应用Chapter 6 Applications of the Definite Integrals (75)6.1 定积分的元素法(The Element Method of Definite Integra (75)6.2 定积分在几何学上的应用(Applications of the DefiniteIntegrals to Geometry) (76)6.3 定积分在物理学上的应用(Applications of the DefiniteIntegrals to Physics) (79)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebar (80)7.1 向量及其线性运算(Vector and Its Linear Operation) (80)7.2 数量积向量积(Dot Product and Cross Product) (86)7.3 曲面及其方程(Surface and Its Equation) (89)7.4 空间曲线及其方程(The Curve in Three-space and Its Equation (91)7.5 平面及其方程（Plane in Space and Its Equation) (93)7.6 空间直线及其方程（Lines in and Their Equations) (95)第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of SeveralVariables and Its Application (99)8.1 多元函数的基本概念（The Basic Concepts of Functionsof Several Variables) (99)8.2 偏导数(Partial Derivative) (102)8.3 全微分(Total Differential) (103)8.4 链式法则（The Chain Rule) (104)8.5 隐函数的求导公式(Derivative Formula for Implicit Functions). (104)8.6 多元函数微分学的几何应用(Geometric Applications of Differentiationof Ffunctions of Severalvariables) (106)8.7方向导数与梯度(Directional Derivatives and Gradients) (107)8.8多元函数的极值(Extreme Value of Functions of Several Variables) (108)第九章重积分Chapter 9 Multiple Integrals (111)9.1二重积分的概念与性质(The Concept of Double Integralsand Its Properities) (111)9.2二重积分的计算法(Evaluation of double Integrals) (114)9.3三重积分(Triple Integrals) (115)9.4重积分的应用(Applications of Multiple Itegrals) (120)第十章曲线积分与曲面积分Chapte 10 Line Integrals and Surface Integrals (121)10.1 对弧长的曲线积分(line Intergrals with Respect to Arc Length) (121)10.2 对坐标的曲线积分(Line Integrals with respect toCoordinate Variables) (123)10.3 格林公式及其应用(Green's Formula and Its Applications) (124)10.4 对面积的曲面积分(Surface Integrals with Respect to Aarea) (126)10.5 对坐标的曲面积分(Surface Integrals with Respect toCoordinate Variables) (128)10.6 高斯公式通量与散度(Gauss's Formula Flux and Divirgence) (130)10.7 斯托克斯公式环流量与旋度(Stokes's Formula Circulationand Rotation) (131)第十一章无穷级数Chapter 11 Infinite Series (133)11.1 常数项级数的概念与性质(The concept and Properties ofThe Constant series) (133)11.2 常数项级数的审敛法(Test for Convergence of the Constant Series) (137)11.3 幂级数(power Series). (143)11.4 函数展开成幂级数(Represent the Function as Power Series) (148)11.5 函数的幂级数展开式的应用(the Appliacation of the Power Seriesrepresentation of a Function) (148)11.6 函数项级数的一致收敛性及一致收敛级数的基本性质(The UnanimousConvergence of the Series of Functions and Its properties) (149)11.7 傅立叶级数（Fourier Series） (152)11.8 一般周期函数的傅立叶级数(Fourier Series of Periodic Functions) (153)第十二章微分方程Chapter 12 Differential Equation (155)12.1微分方程的基本概念（The Concept of DifferentialEquation) (155)12.2可分离变量的微分方程(Separable Differential Equation) (156)12.3齐次方程(Homogeneous Equation) (156)12.4 一次线性微分方程(Linear Differential Equation of theFirst Order) (157)12.5全微分方程(Total Differential Equation) (158)12.6可降阶的高阶微分方程(Higher-order DifferentialEquation Turned to Lower-order DifferentialEquation) (159)12.7高阶线性微分方程(Linear Differential Equation of HigherOrder) (159)12.8常系数齐次线性微分方程(Homogeneous LinearDifferential Equation with Constant Coefficient) (163)12.9常系数非齐次线性微分方程(Non HomogeneousDifferential Equation with Constant Coefficient) (164)12.10 欧拉方程(Euler Equation) (164)12.11 微分方程的幂级数解法(Power Series Solutionto Differential Equation) (164)第三部分常用数学符号的英文表达Part 3 English Expression of the Mathematical Symbol in Common Use第一部分英汉微积分词汇Part1 English-Chinese Calculus V ocabulary 第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood 映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function 单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于 a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数()f x当x趋于x0时的极限limit of functions () f x as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order 等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数()f x在x0连续the function ()f x is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数()f x在该区间上连续function ()f x is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function 导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives()f x在闭区间【a,b】上可导()f x is derivable on the closed interval [a,b]切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error 绝对误差absolute error 相对误差relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term 麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的concave downward’concave down拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) mximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(antiderivative) 积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution 递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector 三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product,dot product,inner product 曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface 椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve 空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes 点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative 混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient 数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values 条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test 交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series 阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system 傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation 常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation 初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order 自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation 串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency 简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficient第二部分定理定义公式的英文表达Part2 English Expression for Theorem, Definition and Formula第一章函数与极限Chapter 1 Function and Limit1.1 映射与函数 (Mapping and Function)一、集合 (Set)二、映射 (Mapping)映射概念 (The Concept of Mapping) 设X , Y 是两个非空集合 , 如果存在一个法则f ，使得对X 中每个元素x ，按法则f ，在Y 中有唯一确定的元素y 与之对应 , 则称f 为从X 到 Y 的映射 , 记作:f X Y →。
托马斯微积分课件7.3 Partial Fractions
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Example 3. Evaluate the indefinite integral
Solution.
x
2 x 4
x
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Example 3. Evaluate the indefinite integral
Solution.
Example 4. Evaluate the indefinite integral
Solution.
x3 x 2 x 1 2 dx x2 x 2 dx x x2 x2 x 1 dx 2 dx x x2
目录
Chapter 7 Integration Techniques, L’Hospital
Rule, and Improper Integrals
7.1 Basic Integration Formulas
7.2 Integration by Parts
7.3 Partial Fractions 7.4 Trigonometric Substitutions 7.5 Integral Tables, CAS, and Monte Carlo Integration 7.6 L’ Hospital Rule
1
2t
2t 1t 2
2 1 t ) 2 (1 2 1t 1t
22 1 t
1 1 dt t 2 2 t
dt
2t 1 t 2
1 t 2
1 1 2 t 2t ln t C 2 2
sin x
cos x
微积分学PPt标准课件23-第23讲微积分的基本公式
确定的I定 bf(积 x)dx分 与值 之 . 对应 a 这意f(味 x)的 着 定b积 f(x)d分 x与它的上 a
之间存在一种函数关系.
固定积分 ,让 下 积 限 分 不 ,上 则 变 限 得变 到
分上限函数:
x
x
F ( x ) a f( x ) d x a f( t) d tx [ a ,b ] .
x
x
由夹逼 x的 定 任 ,即 理 意 F 可 及 (x)性 C 得 (点 a [,b ].)
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8
定理1说明: 定义在区[a间 ,b]上的 积分上限函数是连 . 续的
积分上限函数是否可导?
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9
由 F (x x ) F (x )x xf( t)d t, x
如果 f(x)C(a [,b])则 , 由积分,中 得值定
F ( x ) F ( x x ) F ( x )
x x
x
x x
a f( t) d t a f( t) d t x f( t) d t
又 f( x ) R (a ,[ b ]故 )f ,( x )在 [ a ,b ]上|f有 ( x )| M .界
于 0 | F ( 是 x ) | |x x f ( t ) d t | x x |f ( t ) |d t M x
所以，我们只需讨论积分上限函数.
bf (t)dt 称为积分下限函 . 数 x
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定理 1 若 f ( x ) R ( a , b [ ]则 ) F ( x , ) x f ( t ) d t C ( a , b [ ] .) a 证 x [ a , b ] ,且 x x [ a , b ] ,则
了解利用建立递推关系式求积分的方法.
托马斯微积分课件11.3 Partial derivatives
Example 4. Find the partial derivatives at any point.
yz ln z x y
z z x yz 1
z z 1 z y yz 1
2 z 2 z 2 z yz 3 2 z 3 x y y x yz 1
11.3
Partial Derivatives (偏导数)
Notes：
df x, y0 1. f x x0 , y0 dx x x
0
2.
z f x x0 , y0 , , zx x0 , y0 . x x0 , y0
3.
Notes：
z 1. f y x0 , y0 , , z y x0 , y0 . y x0 , y0
Example 8.
Exercises
P899 14, 18, 21, 22. P900 30, 44, 57, 58.
(R 为常数) , 例6. 已知理想气体的状态方程 求证: p V T 1 V T p RT p RT 2 , 证: p 说明: 此例表明, V V V 偏导数记号是一个 RT V R V , p T p 整体记号, 不能看作 分子与分母的商 !
p V T RT 1 V T p pV
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Example 7.
f x, y x 2 y 2 , 0,0 .
z f x x, y , x
z f y x, y . y
z 2z z 2 z 2 f x x x, y ; f x y x, y y x y x x x x 2 2 z z z z 2 f y y x, y x y f y x x, y ; y y y x y
托马斯微积分课件6.4 First-Order Separable Differential Equations
Differential Equations
6.1 Logarithms
6.2 Exponential Functions
6.3 Derivatives of Inverse Trigonometric Functions; Integrals
Solution. Let
then
and the equation is converted to It is a separable equation. Proceeding as before, we have
Therefore, the solution is given by
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Example 2. Solve the differential equation
Solution.
dy dy 2 3x y 3x 2dx, y 0. dx y目录源自上页下页返回
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Example 3. Solve the following IVP
Solution. Separating the variables, we have
It pass through (1, -1) , thus
Initial condition
Therefore y x3 2.
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Basic conceptions of differential equations
1. Order(阶) 2. Solution(解) 3. Initial Value Problem (IVP,初值问题)
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(定积分换元法)
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Proof. Let F denote any antiderivative of f
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Solution 1.
Solution 2.
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Example. Evaluate the integrals
Solution.
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Example. Evaluate the area of the shade region.
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Example. Evaluate the area of the shade region.
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Exercises
P371 1(a), 4(a), 5(a), 6(a), 8(a), 12, 15. P372 19, 24.
Step 2. Select some number in each subinterval, and then obtain many rectangles.
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Step 3. Take the sum of above products.
Step 4. Take the limitation of Riemann sum.
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Example. Evaluate the integrals
Solution.
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4.6.2
Area between Curves (曲线之间的平面图形面积)
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Step 1. Subdivide the interval [a,b] into subintervals.
Chapter 4 Integration
4.1 Indefinite Integrals, Differential Equations, and Modeling 4.2 Integral Rules, Integration by Substitution 4.3 Estimating with Finite Sums
that both f and g are continuous. Therefore, we have
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Solution.
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Solution 1.
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Solution 2.
Does it has other Solutions?
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4.6
Substitution in Definite Integrals
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4.6.1 Substitution in Definite Integral 4.6.2 Area between Curves
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4.6.1
Substitution in Definite Integral
4.4 Riemann Sums and Definite Integrals 4.5 The Mean Value and Fundamental Theorems
4.6 Substitution in Definite Integrals 4.7 Numerical Integration
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Example. Evaluate the area of the shade region.
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Example. Evaluate the area of the shade region.
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Example. Evaluate the area of the shade region.