7097044-Fractional-Calculus
fractal and fractional佩普学术 -回复
fractal and fractional佩普学术-回复Fractal and Fractional PEP AcademicIntroduction:Fractals and fractions are two mathematical concepts that have significant applications in various fields, including physics, computer graphics, and finance. This article aims to provide a comprehensive understanding of fractals and fractions, exploring their basic definitions, properties, and real-life applications.I. Fractals:1. Definition:Fractals are geometric shapes that exhibit self-similarity, meaning that they contain smaller copies of themselves in a never-ending pattern. They can be generated through a mathematical process called recursion. Examples of well-known fractals include the Mandelbrot set and the Sierpinski triangle.2. Properties:Fractals possess several distinctive properties, including infinite complexity, fractional dimension, and non-integer scaling. These properties contribute to their unique visual appearance and make them applicable in various fields, such as computer graphics and image compression.3. Applications:Fractals find applications in many practical areas. In computer graphics, they are used for creating realistic landscapes, textures, and natural objects. Fractal-based algorithms are also employed in image compression techniques, enabling efficient storage and transmission of digital images. Additionally, fractal analysis is utilized in medical imaging, financial forecasting, and weather prediction.II. Fractions:1. Definition:Fractions are numerical expressions representing a part or parts ofa whole. They consist of a numerator and a denominator, with the numerator representing the number of parts involved and the denominator indicating the total number of equal parts that make up the whole. For example, 3/4 represents three parts out of four equal parts.2. Properties:Fractions possess various properties, including equivalence, addition, subtraction, multiplication, and division. Equivalent fractions represent the same part-to-whole ratio, while adding, subtracting, multiplying, or dividing fractions follow specific rules and algorithms.3. Applications:Fractions have numerous real-life applications. In cooking and baking, fractions are used to determine ingredient quantities accurately. In finances, fractions are utilized to calculate interest rates, percentages, and financial ratios. Moreover, fractions play a significant role in measurements, allowing precise representations of lengths, weights, and volumes.III. Fractals and Fractions:1. Fractional Crystals:Fractional crystals are a special type of fractal pattern that combines the concepts of fractals and fractions. They are formed by repeatedly replacing parts of a shape with smaller copies. Each iteration involves dividing the shape into fractions of the original size and replacing them with smaller-scale copies.2. Applications:Fractional crystals offer an effective way to represent complex structures with fractional dimensions. They find applications in physics, chemistry, and materials science. For instance, they are used to model the behavior of polymers, the structure of porous materials, and the properties of amorphous solids.Conclusion:Fractals and fractions are fundamental mathematical concepts withsignificant practical applications. Fractals exhibit self-similarity and possess unique properties, making them useful in computer graphics, image compression, and numerous scientific fields. Fractions, on the other hand, represent parts of a whole and find applications in cooking, finance, and measurements. The combination of fractals and fractions leads to the concept of fractional crystals, enabling the representation of complex structures with fractional dimensions. Understanding these concepts is essential for anyone interested in mathematics or its various applications.。
The Fundamental Theorem of Calculus
P (x)dx
= (165x − 0.05x2 )|1600 1500 = 1000 . Thus, increasing monthly production from 1,500 units to 1,600 units will increase the monthly profit by $1,000. (See College Mathematics for Business, §12-5, Example 5.)
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
2024年AP Calculus AB真题集锦
2024年AP Calculus AB真题集锦在2024年的AP Calculus AB考试中，考生们将面临一系列挑战，需要全面掌握各个知识点和技巧来应对。
本文将为大家提供一个真题集锦，帮助大家更好地备考。
第一部分：选择题选择题是AP Calculus AB考试中的重要组成部分，考察学生对基础知识的理解和应用能力。
在这个部分，我们将精选几道典型的选择题，供大家练习。
题目1：设函数f(x)为连续函数，且在区间[0,1]上满足f(0)=0，f(1)=2。
已知f'(x)>0，且在[0,1]上有f''(x)<0，则下列哪一个选项是正确的？A. f(x)在(0,1)内取值为负数B. f(x)在(0,1)内取值为正数C. f(x)在(0,1)内取值为0D. 无法确定f(x)在(0,1)内的取值解析：根据题意可知，函数f(x)在[0,1]上是单调递增的，同时具有凹向下的图像形状。
因此，f(x)在(0,1)内取值为正数，选项B为正确答案。
题目2：已知函数f(x)=2x^3-3x^2-12x+5，求f'(2)的值。
A. 12B. 14C. 16D. 18解析：首先，我们对f(x)进行求导，得到f'(x)=6x^2-6x-12。
将x=2代入f'(x)中，得到f'(2)=6(2)^2-6(2)-12=12。
因此，选项A为正确答案。
第二部分：解答题解答题是AP Calculus AB考试中的较为复杂和综合性要求较高的部分，需要考生们综合运用所学知识来解决问题。
下面，我们选择两道典型的解答题供大家参考。
题目3：已知函数f(x)在区间[-1,3]上连续，且满足f'(x)=2(x-1)和f(1)=4。
若曲线y=f(x)与x轴所围成的面积为8，求函数f(x)在区间[-1,3]上的最小值和最大值。
解析：首先，根据题意可以确定f(x)的一阶导数为f'(x)=2(x-1)，再对f'(x)进行积分，得到f(x)=x^2-2x+C，其中C为常数。
AP Calculus AB考试2024历年题目全解
AP Calculus AB考试2024历年题目全解2024年的AP Calculus AB考试历年题目全解[注意：本文以解题步骤为主要内容，为了更好地呈现题目解析，会包含大量的公式和符号。
请确保在阅读时有一定的数学基础。
]1. 第一题题目描述：给定一个函数 f(x)，求其在区间 [a，b] 上的定积分。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用不定积分的方法解决这个问题。
首先，先求 f(x) 的原函数 F(x)，然后计算 F(b) -F(a) 即可得到定积分的结果。
具体步骤如下：1. 求函数 f(x) 的原函数 F(x)。
2. 计算 F(b) - F(a) 的值，即为所求定积分的结果。
2. 第二题题目描述：计算函数 f(x) 的导数。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用求导的方法来计算其导数。
根据导数的定义，导数 f'(x) 可以通过计算 f(x) 的极限来得到。
具体步骤如下：1. 对函数 f(x) 求导。
2. 化简导数的表达式，得到最终结果。
3. 第三题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的极限。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用极限的定义来计算其在某点 x=c 处的极限。
根据极限的定义，我们需要分别计算 x 无限接近 c 时 f(x) 的左极限和右极限。
具体步骤如下：1. 计算 f(x) 在 c 的左极限。
2. 计算 f(x) 在 c 的右极限。
3. 判断左极限和右极限是否存在且相等，若相等则极限存在，否则不存在。
4. 第四题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的导数。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用导数的定义来计算其在某点 x=c 处的导数。
导数 f'(x) 可以通过计算 f(x) 的极限来得到。
具体步骤如下：1. 对函数 f(x) 求导。
2. 将 x 的值替换为 c，计算导数的值，即为所求导数的结果。
2012国际Caculus(I)期中(上)考试卷-副本(2)
江西财经大学2012级期中（上）考试试卷课程代码：12003（A ） 授课课时：48 考试时间：110分钟 课程名称：Calculus (I) 适用对象：12级国际学院本科生 试卷命题人： 聂 高 辉 . 试卷审核人： 聂 高 辉 .1. Full in the blank of each statement in the following five statements such that the statement is right. Then write the corresponding answer on the answer book by the title number. You can be gained 3 points for the right thing on per blank.(1) (sec )x '=________.(2) 0sin 2limx x x→=_________. (3) d (log )d a x x =________. (4) 2lim 1x x x →∞⎛⎫+= ⎪⎝⎭________.(5) The tangent line equation of the cure ()f x =(1,1)is________.2. Choose the one that the statement is right from four choices marked A, B, C, and D, with which each statement of the following five statements. Then write the corresponding answer on the answer book by the title number. You can be gained 3 points for the right choice of per the statement.(6)1d |x z ==________provided that 2z x =(A) d x . (B) 2d x . (C) 0. (D) 2. (7) If (1)2f =, (1)6f '=and (2)8f '=, then 12d ()|d x f x x-== . (A) 1/8. (B) 1/6. (C) 1/2. (D) 1.(8) Differentiable functions ()f x and ()g x with ()()[,]f x g x x a b <∈hold property for which .(A) ()()[,]f x g x x a b ''<∈ (B) ()()[,]f x g x x a b ''≤∈(C) 000lim ()lim ()(,)x x x x f x g x x a b →→≤∈ (D)000lim ()lim ()(,)x x x x f x g x x a b →→<∈ (9) If sin lim x x a x →∞=and 01lim sin x x b x→=, then (,)a b =________ (A).(1,1). (B). (1,0). (C).(0,1). (D) (0,0).(10) If 0()2f x =, then in the following propositions is right.(A)0lim ()2x x f x →=. (B)0()0f x '=. (C)0lim ()2x x f x →≠. (D)0d (())0d f x x=. 3. Give the solution of any of in the following problems and write operation process and answer on the answer book by the title number. You can be gained mark of per problem by the right process and answer.(11) (10pts) Find 21x →and 2lim()x x →∞ (12) (6pts) 210lim(sec )x x x → (13) (6pts) Evaluate 2lim sin 1x x x x →∞+(14) (8pts). Find d d y x if sin sin x x y x x =(15) (6pts). Compute x + (16) (8pts) Find d d y x , 22d d x yif xy x y e += (17) (8pts) Find d d n n z xif (21)ln(21)z x x =++ (17). (12pts) A company estimates that the demand function of an item on the market is ()2003D p p =-.(1) Find marginal demand at 2p =and interpret its meaning? (2)If the formula ()()D p p D p 'is said to be elasticity of demand with respective to price at p , then what is the elasticity of demand of this item at 2p =? Can you say meaning of the elasticity?(3)Please you give the elasticity formula of a general function ()f x .(18) (6pts) Prove that )(x f is continuous at 0x if and only if 00lim[()()]0x x f x f x →-=。
AP Calculus AB历年真题精选2024
AP Calculus AB历年真题精选2024 2024年的AP Calculus AB考试即将到来，对于准备参加考试的学生来说，熟悉历年真题是非常重要的。
在本文中，我将为大家精选几道2024年AP Calculus AB考试的历年真题，帮助大家更好地备考。
一、选择题1. 求函数f(x) = x^3 - 3x^2 + 2x的导数。
解析：根据导数的定义，对函数f(x)求导数，我们可以将每一项的指数乘以系数，并减少指数。
因此，f'(x) = 3x^2 - 6x + 2。
2. 设函数g(x) = (x - 2)^2 + 3，求g(x)的最小值。
解析：由于(x - 2)^2的值永远大于等于0，所以g(x)的最小值就是当(x - 2)^2的值为0时，即x = 2。
代入函数g(x)，可以得到最小值为3。
二、计算题1. 求函数y = x^2 + 4x + 3的定积分，区间为[-1, 2]。
解析：我们可以先对函数y = x^2 + 4x + 3求不定积分，得到原函数F(x) = (1/3)x^3 + 2x^2 + 3x + C。
然后，我们将上下限带入原函数，得到积分值为F(2) - F(-1) = 11。
2. 求曲线y = x^3在x = 2处的切线方程。
解析：首先，我们需要求得曲线y = x^3的导数，即y' = 3x^2。
然后，代入x = 2，得到斜率k = 12。
由于切线方程的一般形式为y = kx +b，我们可以代入已知的点(x, y) = (2, 8)，解得b = -4。
因此，切线方程为y = 12x - 4。
三、证明题证明：对于任意实数x和y，有(x + y)^2 ≥ 4xy。
解析：我们可以展开(x + y)^2，得到x^2 + 2xy + y^2。
根据平方项的非负性，我们可以得出x^2 ≥ 0和y^2 ≥ 0。
因此，x^2 + y^2 ≥ 0，进而有x^2 + y^2 + 2xy ≥ 2xy。
AP_calculus参考书
Score
AP Score 5 4 3 2 1 Qualification Extremely well qualified Well qualified Qualified Possibly qualified No recommendation
Topic Outline for Calculus BC
1
by taking sufficiently close values of the domain.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). * Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. II. Derivatives Concept of the derivative • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. • Instantaneous rate of change as the limit of average rate of change. • Approximate rate of change from graphs and tables of values. Derivative as a function • Corresponding characteristics of graphs of ƒ and ƒ∙. • Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’. • The Mean Value Theorem and its geometric interpretation. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Second derivatives • Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’∙. • Relationship between the concavity of ƒ and the sign of ƒ’∙. • Points of inflection as places where concavity changes. Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration. • Optimization, both absolute (global) and relative (local) extrema. • Modeling rates of change, including related rates problems. • Use of implicit differentiation to find the derivative of an inverse function. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. + Numerical solution of differential equations using Euler’s method. + L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series. Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
第二单元参考书目-中国科学院数学与系统科学研究院
现代微分几何
李洪波、刘卓军
《微分几何讲义》，陈省身，陈维桓著。
纠错码理论
万哲先、黄民强、邓映蒲、 《代数和编码》万哲先著，科学出版社。 F.J.MacWillams and N.J.A. Sloane, “The Theory of Error
刘卓军
–Correcting Codes (PartⅠ)” 。
第三单元参考书目
学科专业
第三单元
（二级学科）
（专业课）
代数表示论
李代数
代数拓扑
数论
代数几何
偏微分方程（乙）
动力系统 系统与控制理论
数学物理
基础数学 复动力系统与值分布
论
多复变与复几何
动力系统 偏微分方程（乙）
算子代数
偏微分方程
数论 常微分方程
导师
考试覆盖内容及参考书
韩阳
席南华、赵开明、徐晓平
段海豹、潘建中
高全泉
同运筹学与控制论 同系统建模与控制 《人工智能》(上、下册)，陆汝钤著，科学出版社。 Roger S. Pressman, Software Engineering A Practitioner’s Approach (软件工程：实践者的研究方法)， R. S. Pressman & Associates 公司，机械工业出版社。
学出版社, 2005), Chapters 11 & 19
杨乐、王跃飞、崔贵珍
周向宇
贺正需 张平
葛力明
丁彦恒 张志涛 葛力明 郑作环
F.Warner: Foundations of differential manifolds and Lie groups, GTM94。 Griffiths,Harris: Principles of algebraic geometry. John Wiley & Sons, Inc., 1978; Fritzsche, Grauert: From holomorphic functions to complex manifolds. GTM213, Springer, 2002。
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1.2 Historical Survey Most authors on this topic will cite a particular date as the birthday of so called 'Fractional Calculus'. In a letter dated September 30th, 1695 L’Hospital wrote to Leibniz asking him about a particular notation he had used in his publications for the nth-derivative of the Dn x linear function f(x) = x, . L’Hospital posed the question to Leibniz, Dx n what would the result be if n = 1/ 2. Leibniz's response: "An apparent paradox, from which one day useful consequences will be drawn." In these words fractional calculus was born. [7]
Chapter 1
Definition and Applications
3
Fractional Calculus History
value x, verifiable by infinite series expansion, or more practically, by calculator. Now, in the same way consider the integral and derivative. Although they are indeed concepts of a higher complexity by nature, it is still fairly easy to physically represent their meaning. Once mastered, the idea of completing numerous of these operations, integrations or differentiations follows naturally. Given the satisfaction of a very few restrictions (e.g. function continuity) completing n integrations can become as methodical as multiplication. [7] But the curious mind can not be restrained from asking the question what if n were not restricted to an integer value? Again, at first glance, the physical meaning can become convoluted (pun intended), but as this report will show, fractional calculus flows quite naturally from our traditional definitions. And just as fractional exponents such as the square root may find their way into innumerable equations and applications, it will become apparent that integrations of order 1/ 2 and beyond can find practical use in many modern problems. [7]
and the integration operator J. (Usually J is used in favor of I to avoid confusion with other I-like glyphs and identities) [1] Fractional calculus comes out of generalizing the differential/integral operator, applying it at non-integer order (example, a "half derivative" operation such that taking the half derivative of a function twice yields the first derivative The traditional integral and derivative are, to say the least, a staple for the technology professional, essential as a means of understanding and working with natural and artificial systems. [7] Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. [7]
Chapter 1
Definition and Applications
2
Fractional Calculus History
1.1
Definition
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operators
Chapter 1
Definition and Applilus History
Leibniz
Following L’Hospital and Leibniz’s first inquisition, fractional calculus was primarily a study reserved for the best minds in mathematics. Fourier, Euler, Laplace are among the many that dabbled with fractional calculus and the mathematical consequences. Many found, using their own notation and methodology, definitions that fit the concept of a non-integer order integral or derivative. [7] The most famous of these definitions that have been popularized in the world of fractional calculus (not yet the world as a whole) are the Riemann-Liouville and Grunwald-Letnikov definition. While the shear numbers of actual Definitions are no doubt as numerous as the men and women that study this field, they are for the most part variations on the themes of these two and so are addressed in detail in this document. [7] Most of the mathematical theory applicable to the study of fractional calculus was developed prior to the turn of the 20th century. However it is in the past 100 years that the most intriguing leaps in engineering and scientific application have been found. The
Chapter 1
Definition and Applications
5
Fractional Calculus History
mathematics has in some cases had to change to meet the requirements of physical reality. [7] Caputo reformulated the more 'classic' definition of the RiemannLiouville fractional derivative in order to use integer order initial conditions to solve his fractional order differential equations. As recently as 1996, Kolowankar reformulated again, the RiemannLiouville fractional derivative in order to differentiate no-where differentiable fractal functions. [7] Leibniz's response, based on studies over the intervening 300 years, has proven at least half right. It is clear that within the 20th century especially numerous applications and physical manifestations of fractional calculus have been found. However, these applications and the mathematical background surrounding fractional calculus are far from paradoxical. While the physical meaning is difficult (arguably impossible) to grasp, the definitions themselves are no more rigorous than those of their integer order counterparts. [7] Understanding of definitions and use of fractional calculus will be made clearer by quickly discussing some necessary but relatively simple mathematical definitions that will arise in the study of these concepts. These are The Gamma Function, The Beta Function, The Laplace Transform, and the Mittag-Leffler Function and are addressed in the second and third chapter of this thesis. [7]