Exponential and logarithmic functions

合集下载

ap-预备微积分

ap-预备微积分AP Precalculus 预备微积分为什么要开设AP Precalculus？如果高中期间没有做好充分的准备，对于很多大学新生来说，入门级数学的学习还是比较吃力的，学生们很难完成大学数学入门级课程，同时能够获得的学习支持也很少。

为了帮助学生们能够顺利完成大学数学的入门过渡，帮助更多高中生尽早夯实数学基础，CB便考虑开设AP Precaculus课程，即“微积分预备”。

简而言之，AP Precalculus开设可以归类为以下几类好处：▪准备：为学生更好地学习大学数学提供准备；▪鼓励：鼓励更多学生在高中完成四年的数学学习；▪培养：帮助想要就读STEM的学生完成目标；▪装备：让学生具备完成大学学业要求所需的数学技能；AP Precalculus 课程内容课程内容Unit 1: Polynomial and Rational FunctionsUnit 2: Exponential and Logarithmic Functions Unit 3: Trigonometric and Polar Functions Unit 4: Functions Involving Parameters, Vectors, and Matrices每个单元都是未来大学数学科目的重要基础。

通过AP Precalculus 课程的学习，学生们在大学的数学基础课的学习中更能从容应对。

主要培养学生的以下3项技能：•Procedural and Symbolic Fluency •Multiple Representations •Communication and ReasoningAP Precalculus 课程大纲AP Precalculus 考试形式AP Precalculus的考试和AP Calculus整体比较相似，分为选择题和简答题两部分。

值得注意的是，这次新加入的AP Precalculus同样分为计算器部分和非计算器部分，说明对于学生的计算能力还是有一定考察的。

自然对数函数

假設將本金 P0 存放於銀行，年利率為 r 且每年複利 k 次，即每 365/k 天複利一次。在這種情況下，每次複利之利率為 r/k 而一年後之本利和為
n 年後之本利和為 P(1)P0(1kr)k P(n)P0(1kr)nk
4-1 指數函數
求本利和
將1000元存放於銀行，年利率為6%且每年複利一次，試問5 年後之本利和為多少？ P(5)10(0 100.0)651338
將1000元存放於銀行且銀行之年利率為8%。
➢ (a) 每年複利一次，試問2年後之本利和為多少？
P(2)10(0 100.0)821166
➢ (b) 每半年複利一次，試問2年後之本利和為多少？
P (2 ) 10 (1 0 0 .2 0) 0 4 8 10 (1 0 0 .00 )4 4 1170
(d) f(x) 為連續函數。
(e) 若 a > 1，則 f(x) 為遞增函數， limf(x)， lim f(x)0 ，
其圖形如左圖所示。
x
x-
(f) 若 0 < a < 1，則 f(x) 為遞減函數， limf(x)0，limf(x)，
其圖形如右圖所示。
x
x-
4-1 指數函數
複利問題
最典型的指數函數的例子，即所謂的複利 (compounded
x
1x)x
是否存在？
自然指數
定義4-2: exl i m(11x)x2.7182稱8為自然指數 (natural exponent)。
定義4-3: 連續複利(continuously compounded interest) 將本金 P0 元存於年利率 r 的銀行裡，在連續複利之下，t 年後之本利和為 P(t) = P0ert。

log指数函数运算法则

log指数函数运算法则Logarithmic and Exponential Function OperationsLogarithmic and exponential functions are important mathematical concepts that are used in various fields such as finance, science, and engineering. Understanding the rules for operating with these functions is essential for solving problems involving them. In this article, we will discuss the rules for operating with logarithmic and exponential functions.Logarithmic Function OperationsThe following are the rules for operating with logarithmic functions:1. Product Rule: log a (mn) = log a m + log a nThis rule states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. For example, log2(4x) = log2(4) + log2(x) = 2 + log2(x).2. Quotient Rule: log a (m/n) = log a m - log a nThis rule states that the logarithm of the quotient of twonumbers is equal to the difference of the logarithms of the individual numbers. For example, log3(27/9) = log3(27) - log3(9) = 3 - 2 = 1.3. Power Rule: log a mn = n log a mThis rule states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. For example, log2(3^4) = 4 log2(3) = 4 × 1.585 = 6.34.4. Change of Base Rule: log a b = log c b / log c aThis rule states that the logarithm of a number to a given base can be expressed as the logarithm of the same number to a different base divided by the logarithm of the original base to the same different base. For example, log3(5) = log10(5) / log10(3) = 0.699 / 0.477 = 1.46. Exponential Function OperationsThe following are the rules for operating with exponential functions:1. Product Rule: a^m × a^n = a^(m+n)This rule states that the product of two exponential expressions with the same base is equal to an exponential expression with the same base raised to the sum of the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.2. Quotient Rule: a^m / a^n = a^(m-n)This rule states that the quotient of two exponential expressions with the same base is equal to an exponential expression with the same base raised to the difference of the exponents. For example, 5^7 / 5^4 = 5^(7-4) = 5^3.3. Power Rule: (a^m)^n = a^(mn)This rule states that an exponential expression raised to a power is equal to the base raised to the product of the power and the exponent. For example, (4^3)^2 = 4^(3×2) = 4^6.4. Negative Exponent Rule: a^-m = 1 / a^mThis rule states that an exponential expression with a negative exponent is equal to the reciprocal of the same expression with a positive exponent. For example, 2^-3 = 1 / 2^3 = 1 / 8.ConclusionIn conclusion, logarithmic and exponential function operations are important concepts in mathematics. The rules for operating with these functions are essential for solving problems involving them. By understanding these rules, you can simplify complex expressions and solve equations involving logarithmic and exponential functions.。

对数的定义

第 4 章
指数函数和对数函数
Exponential Function and Logarithmic Function
对数
温故知新课前思考新课讲解课堂练习
引入
返回主页
假设某厂的生产总值今年为a万元，如果每年平均以10%增加，那么该厂经ห้องสมุดไป่ตู้几年的努力，生产总值为今年的2倍？
解：设经过x年后，该厂的生产总值 2a万元达到
a
loga M
M
7
课堂练习 1
1、用log a x , log a y, log a z表示下列各式
2
3 4 5
xy (1) log a z
3
log a x log a y log a z
2 3
( 2) log a x y z 3 log a x 2 log a y 3 log a z
2
3 4 5
解：原式 log 1 2
2
1 log 1 2 2 1 2
1 2
lg 2 原式 1 lg 2 1 lg 2 2 lg 2 1 2
1 2
课堂练习 1
2
3 4 5
1.用换底公式或计算器计精确到0.001 算 lg 5 1 log 3 5 1.465 lg 3 2 log 1 3 lg 3 0.683 5 lg 5 3 log 3 0.23 lg 0.23 1.338 lg 3 lg 5.2 4log 0.3 5.2 1.369 lg 0.3
根据题意：即：
a 1 10% 2a
x
1.1 2
x
已知底数和幂的值，求指数的问题，这就是我们要学的对数问题。

不定积分的概念教案

不定积分的概念教案Lesson Plan on the Concept of Indefinite Integral教学目标：1.了解不定积分的基本概念及意义。

2.掌握不定积分的符号表示和性质。

3.学会计算基本的不定积分。

教学内容：Introduction:In this lesson, we will introduce the concept of indefinite integral and understand its significance.We will also explore the notation and properties of indefinite integrals.引入：本节课我们将介绍不定积分的基本概念及其意义。

我们将探讨不定积分的符号表示和性质。

Section 1: Definition and Significance of Indefinite Integral1.1 Definition:An indefinite integral of a function f(x) is a function whose derivative is f(x), and it is denoted by ∫f(x)dx.The process of finding an indefinite integral is called antiderivative.1.2 Significance:Indefinite integrals play a crucial role in calculus.They are used tosolve problems involving area, volume, and accumulation.They also provide the foundation for calculating definite integrals, which are used to find exact values of functions.1.1 定义：函数f(x)的不定积分是一个导数为f(x)的函数，用符号∫f(x)dx表示。

六种基本初等函数（elementaryfunction）

六种基本初等函数（elementary function）
一、常数函数（constant function）
因为f映射任意的值到4，因此函数f(x)是一个常数。

二、幂函数（power function)
形如y=x^a(a为常数）的函数。

如，y = x^ 1/2，y = x，y= x^ 2，y= x^3 等。

三、指数函数(exponential function)
形如y=a^x的函数，式中a为不等于1的正常数。

四、对数函数(logarithmic function)
指数函数的反函数，记作y=loga x式中a为不等于1的正常数，定义域是X〉0。

对数函数图形对数函数与指数函数互为反函数
五、三角函数(trigonometric function)
即正弦函数y=sinx ，余弦函数y=cosx ，正切函数y=tanx，余切函数y=cotx ，正割函数y=secx，余割函数y=cscx。

六、反三角函数(inverse trigonometic function)
反正弦函数y = arcsin x，为y=sin x的反函数反余弦函数y = arccos x，为y=cos x 的反函数
反正切函数y = arctan x，为y=tan x 的反函数反余切函数y = arccot x ，为y=cot x的反函数
反正割函数y = arcsec x ，为y=sec x的反函数反余割函数y = arccsc x ，为y=csc x的反函数七、定义域，值域和单调性。

大一数学知识点公式总结

大一数学知识点公式总结在大一学习数学的过程中，我们会遇到各种各样的知识点和公式。

这些知识点和公式对于数学的学习和理解起着至关重要的作用。

本文将对大一数学中常见的知识点和公式进行总结，帮助大家更好地掌握数学。

1. 代数与函数1.1 复数（Complex Numbers）：- 复数的定义：z = a + bi，其中a为实部，bi为虚部，i为虚数单位。

- 欧拉公式：e^(ix) = cos(x) + i*sin(x)。

1.2 幂函数和指数函数（Power and Exponential Functions）：- 幂函数的性质：a^m * a^n = a^(m+n)。

- 指数函数的性质：a^m / a^n = a^(m-n)。

1.3 对数函数（Logarithmic Functions）：- 对数函数的性质：log_ab = (log_cb) / (log_ca)。

2. 微积分2.1 极限与连续性（Limits and Continuity）：- 极限的定义：lim(x→a) f(x) = L，表示当x趋近于a时，f(x)趋近于L。

- 连续函数的定义：函数f在点a处连续，当且仅当lim(x→a) f(x) = f(a)。

2.2 导数和微分（Derivatives and Differentials）：- 导数的定义：f'(x) = lim(h→0) [f(x+h) - f(x)] / h。

- 常见的导数公式：- (k)' = 0，其中k为常数。

- (x^n)' = n*x^(n-1)，其中n为常数。

- (sin(x))' = cos(x)，(cos(x))' = -sin(x)。

- 微分的定义：df(x) = f'(x) * dx。

2.3 积分（Integrals）：- 不定积分：∫f(x)dx，表示求函数f的原函数。

- 定积分：∫[a,b]f(x)dx，表示求函数f在[a,b]上的面积或曲线长度。

logarithmic function

细胞分裂问题：一个细胞能分裂成两个，两个能分裂成四个，以此类推，那么分裂成64个需要分裂几次？
假设1995年我国国民生产总值为a亿元，如果每年平均增长8%，那么一年后，生产总值是多少？两年呢？那么经过多少年后生产总值是 1995年的2倍？
一年
又一年
.....经x年
a
a(1+8%) a(1+8%)2 1.08x=2
能否得到
如果 a 0, 且a 1 ，M 0, N 0, 那么
(1) loga ( M N ) loga M loga N M (2) loga loga M loga N N n (3) loga M n loga M
对数运算的三条运算法则：
如果 a 0, 且a 1 ，M 0, N 0, 那么
自然对数loge 10记为Ln 10.
上一页
exponential form >>>>>>>logarithmic form
• （1）2－1＝ • （2）27
1 ＝
1 2
3
1 3
上一页
logarithmic form >>>>>>exponential form
Log2
1 =-2 4
Log3
证明：令p loga b，则
a p b, 两边取以c为底的对数, 有 logc a p logc b, 则 p logc a logc b, logc b 所以 p , logc a logc b 即 loga b . logc a
例7 利用换底公式可得: log3 3 1 log2 3 log3 2 log3 2 请利用同样的方法证明:

高一数学函数图像英语阅读理解30题

高一数学函数图像英语阅读理解30题1<背景文章>Linear functions are an important part of high school mathematics. A linear function is represented by the equation y = mx + b, where m is the slope and b is the y-intercept.The slope of a linear function determines the steepness of the line. If the slope is positive, the line goes up as you move from left to right. If the slope is negative, the line goes down. A slope of zero means the line is horizontal.The y-intercept is the point where the line crosses the y-axis. It gives the value of y when x = 0.Linear functions can be used to model many real-life situations. For example, the cost of renting a car may be a linear function of the number of days you rent it. The distance traveled by a car moving at a constant speed is also a linear function of time.Now let's look at an example of a linear function graph. Consider the function y = 2x + 3. The slope of this function is 2, which means for every increase of 1 in x, y increases by 2. The y-intercept is 3, so the line crosses the y-axis at the point (0, 3).1. The equation of a linear function is y = mx + b. Here, m represents___.A. the x-interceptB. the slopeC. the y-interceptD. the value of y答案：B。

二阶连续可微函数英译 -回复

二阶连续可微函数英译-回复Title: Second-Order Continuous Differentiable FunctionsIntroduction:In mathematics, functions are one of the fundamental concepts. They describe the relationship between two sets of numbers, known as the domain and the range. Functions come in various forms, and their properties can be studied to gain a deeper understanding of their behaviors. One important type of function is the second-order continuous differentiable function. This article will delve into the topic, exploring its definition, properties, and applications.I. Definition and Properties:A second-order continuous differentiable function, also known as a twice-differentiable function, is a function that has a continuous first derivative and a continuous second derivative. This implies that the function can be differentiated twice, and these derivatives exist and are continuous over the entire domain.The continuity of the first derivative ensures that the function does not have any abrupt changes or jumps, while the continuity of thesecond derivative indicates how smoothly the function curves and changes direction. These properties make second-order continuous differentiable functions particularly useful in various areas of mathematics, including calculus, optimization, and physics.II. Examples:To understand the concept better, let's consider a few examples of second-order continuous differentiable functions:1. Quadratic Functions:Quadratic functions, such as f(x) = ax^2 + bx + c, where a, b, and c are constants, are second-order continuous differentiable functions. These functions form a family of parabolas that have a single global minimum or maximum point, depending on the leading coefficient.2. Trigonometric Functions:Trigonometric functions, such as sine and cosine, are alsosecond-order continuous differentiable functions. These functions exhibit periodic behavior and are widely used in physics, engineering, and signal processing.3. Exponential and Logarithmic Functions:Exponential and logarithmic functions, such as f(x) = e^x and f(x) = ln(x), respectively, are also examples of second-order continuous differentiable functions. These functions have many applications in growth and decay processes, finance, and probability theory.III. Applications:Second-order continuous differentiable functions play a crucial role in various fields. Here are a few applications where their properties are extensively utilized:1. Optimization:Optimization problems involve finding the maximum or minimum of a given function. Second-order continuous differentiable functions are particularly useful here because of their smoothness and well-defined curvature. Techniques like Newton's method and gradient descent use these functions to efficiently find optimal solutions.2. Physics:Many physical phenomena can be accurately described using second-order continuous differentiable functions. These functions help determine rates of change, acceleration, and other importantquantities in mechanics, electromagnetism, and thermodynamics.3. Regression Analysis:Regression analysis is a statistical method used to model the relationship between variables. Second-order continuous differentiable functions are often employed in regression models because they can accurately capture intricate nonlinear relationships between variables.4. Control Systems:Control systems are widely used in engineering to regulate and stabilize various processes. Second-order continuous differentiable functions play a crucial role in designing controllers and modeling system dynamics, ensuring smooth and precise control.Conclusion:Second-order continuous differentiable functions are a fundamental class of mathematical functions that possess continuous first and second derivatives. These functions provide insights into the smoothness, curvature, and behavior of mathematical models. Their properties make them invaluable invarious fields, including optimization, physics, statistics, and engineering. Understanding and utilizing these functions contribute to the advancement of mathematics and its applications in the real world.。

Chapter 9 Exponential and Logarithmic Functions指数函数和对数函数的导数

Rule No.1: If
y ln x
, then
dy 1 dx x
This rule has a corollary that incorporates the Chain Rule and is actually a more useful rule to memorize: • Rule No.2: If
ln x3 3 ln x
1 3 f x 3 x x

Therefore,
Example 2: Find the derivative of f x ln 5x 3x 6 .
5 18 x f x 5x 3x
ln x log a x ln a
Review the unit on Prerequisite Mathemactics if this leaves you scratching your head. Anyway, because ln a is a constant, we can take the derivative and we get:
Chapter 9 Exponential and Logarithmic Functions, Part One
AP Math Bella
As with trigonometric functions, you'll be expected to remember all of the logarithmic and exponential functions you've studied in the past. If you're not sure about any of this stuff. review the unit on Prerequisite Mathematics. Also, this is only part one of our treatment of exponents and logs. Much of what you need to know about these functions requires knowledge of integrals( the second half of the book), so we'll discuss them again later.

对数不等式的推论证明

对数不等式的推论证明Logarithmic inequalities are a fundamental concept in mathematics that involve the comparison of logarithmic expressions. These inequalities play a crucial role in various fields, including calculus, algebra, and analysis.In this discussion, we will explore the implications and proofs of logarithmic inequalities.One of the most commonly encountered logarithmic inequalities is the natural logarithm inequality, which states that for any positive real numbers a and b, if a > b, then ln(a) > ln(b). This inequality can be easily proven using the properties of logarithms. The natural logarithm function is strictly increasing, meaning that as the input increases, the output also increases. Therefore, if a > b, then ln(a) > ln(b).Another important logarithmic inequality is the exponential inequality, which states that for any positive real numbers a and b, if a > b, then a^x > b^x for any realnumber x. This inequality can be derived from the properties of exponential and logarithmic functions. By taking the logarithm of both sides of the inequality, we obtain xln(a) > xln(b). Since x is a positive real number, we can divide both sides by x without changing the inequality. Thus, we have ln(a) > ln(b), which implies that a^x > b^x.Logarithmic inequalities also find application in solving exponential equations. For example, consider the equation 2^x = 10. By taking the logarithm of both sides, we obtain xln(2) = ln(10). Since ln(2) and ln(10) are both positive real numbers, we can divide both sides by ln(2) to solve for x. This process relies on the fact that the natural logarithm function is strictly increasing, ensuring that the inequality is preserved during the division.In calculus, logarithmic inequalities are used to analyze the behavior of functions. For instance, the inequality ln(x) < x for all x > 0 is often employed to prove the convergence of series and integrals. By comparing the terms of a series or the integrand of an integral to alogarithmic function, we can determine whether the series or integral converges or diverges.In summary, logarithmic inequalities are essential tools in mathematics that allow for the comparison of logarithmic expressions. These inequalities can be proven using the properties of logarithmic and exponential functions and find applications in various mathematical fields. Understanding and utilizing logarithmic inequalities is crucial for solving equations, analyzing functions, and making mathematical conclusions.。

高等数学教材答案下册英语

高等数学教材答案下册英语Unit 1: Functions and Their GraphsChapter 1: Linear Functions1.1 Functions and Their Representations1.2 Domain and Range1.3 Linear Functions and EquationsChapter 2: Quadratic Functions2.1 Graphs of Quadratic Functions2.2 Solving Quadratic Equations2.3 Quadratic Functions and Their Transformations Chapter 3: Exponential and Logarithmic Functions3.1 Exponential Functions and Their Graphs3.2 Logarithmic Functions and Their Graphs3.3 Exponential and Logarithmic EquationsUnit 2: Limits and ContinuityChapter 4: Limits and Continuity4.1 Limits and Their Properties4.2 Continuity and Its Properties4.3 Computing LimitsChapter 5: Derivatives5.1 The Derivative and its Interpretation5.2 Differentiation Techniques5.3 Applications of DerivativesChapter 6: Higher-Order Derivatives6.1 Higher-Order Derivatives and Their Interpretations 6.2 The Chain Rule6.3 Implicit DifferentiationUnit 3: IntegrationChapter 7: Antiderivatives and Indefinite Integrals 7.1 Antiderivatives and Their Properties7.2 Indefinite Integrals7.3 Substitution MethodChapter 8: Definite Integrals and Their Applications 8.1 Definite Integrals and Their Properties8.2 Applications of Definite Integrals8.3 Numerical IntegrationChapter 9: Techniques of Integration9.1 Integration by Parts9.2 Trigonometric Integrals9.3 Trigonometric SubstitutionUnit 4: Differential Equations and Applications Chapter 10: First-Order Differential Equations 10.1 Separable Differential Equations10.2 Linear Differential Equations10.3 Applications of Differential Equations Chapter 11: Applications of Differential Calculus 11.1 Optimization11.2 Related Rates11.3 Newton's MethodChapter 12: Sequences and Series12.1 Sequences and Their Limits12.2 Infinite Series12.3 Convergence TestsUnit 5: Multivariable CalculusChapter 13: Functions of Several Variables 13.1 Functions of Two or More Variables13.2 Partial Derivatives13.3 Optimization of Functions of Two VariablesChapter 14: Multiple Integrals14.1 Double Integrals14.2 Triple Integrals14.3 Applications of Multiple IntegralsChapter 15: Vector Calculus15.1 Vector Fields15.2 Line Integrals15.3 Green's TheoremChapter 16: Differential Calculus of Vector Fields16.1 Gradient Fields and Potential Functions16.2 Divergence and Curl16.3 Stokes' TheoremI hope the above chapters and sections provide a comprehensive overview of the answers to the exercises and problems in the textbook. Remember to utilize this answer key as a useful tool to check your understanding and progress in studying advanced mathematics.。

《科技英语写作》词汇表解析

Glossary物质值函数方程式方程组逻辑方程解方程表达式展开式正极负极电路电流恒量变量电动机发电机图像处理热力学原子原子核分子离子中子质子(章)节无线电波光波力(力的)大小和方向磁电磁学放射性电流电压电阻万用表压力计性能用户数据热量体积自由电子晶体保证输入输出椭圆半圆圆原点直径半径角度交流电直流电参数半导体元素易燃的气体液体气体固体X轴波长阻力定律定理代数几何电子工程师原子能核能电磁波挤压/压缩均匀的薄膜恒定温度上升下降起重机导线电荷码电阻率测量组合呈指数地（变化）十的平方十的立方立方米临界温度垂直线垂直常用符号延时电路图方框图永久地电容器储存电荷二端口网络传播延时显像管公式成正比成反比带宽幂有理数无理数磁场天然的和人造的直线曲线镜头二进制十进制波形摩擦探测雷达发射机防水光谱变压器加减乘除比率比值，分数电子望远镜光学望远镜优越性模-数转换器混合参数功动能万有引力位能气体膨胀核裂变接收机放大器编码系统天线隧穿调制信号数据处理平移运动垂直运动进行试验通讯卫星采样样品机械能电能弹性导数表示幻灯机标准螺钉右旋串联并联动力学酶中暑画线图解(火箭)推力无限无数无限大无限小有限线圈皮带分析模型地磁场金属导体微积分指令细胞膜周期波形循环系统人体组织电流反馈容器新奇事绝缘体漏电真空管倒数浏览器磁感应装置电感器误差罗盘指针集电极晶体管扩散率变化情况导电率概率斜率实用性绝对值极限值Glossary物质substance, material值value函数function方程式equation方程组a set of equations逻辑方程logic equation解方程solve the equation表达式expression展开式expansion正极positive terminal负极negative terminal电路electric circuit电流current, electric current恒量constant变量variable电动机electric motor发电机generator图像处理image processing热力学thermodynamics原子atom原子核nucleus分子molecule离子ion中子neutron质子proton(章)节section无线电波radio wave光波light wave力force(力的)大小和方向magnitude and direction of the force 磁magnetism电磁学electromagnetism放射性radioactivity电压voltage电阻resistance万用表multimeter压力计pressure gauge性能performance用户数据user data热量heat质量mass体积volume自由电子free electron晶体crystal保证guarantee输入input输出output椭圆ellipse半圆semicircular圆circle原点origin直径diameter半径radius, semidiameter角度angle交流电AC alternating current 直流电DC direct current参数parameter半导体semiconductor元素element易燃的气体combustible gas 液体liquid气体gas固体solidX轴x-axis波长wavelength阻力resistance定律law定理theorem代数algebra几何geometry电子工程师electrical engineer 原子能atomic energy核能nuclear energy电磁波electromagnetic wave 挤压/压缩compress均匀的薄膜uniform film恒定温度uniform temperature 上升increase, rise下降decrease, fall起重机crane导线wire电荷charge电阻率resistivity, resistance加速度acceleration测量measure组合combination呈指数地（变化）exponentially十的平方the square of ten十的立方the cube of ten立方米cubic meter临界温度critical temperature垂直线vertical line，垂直perpendicular to常用符号usual symbol延时delay电路图diagram方框图block diagram永久地permanently电容器capacitor储存电荷store charge二端口网络two-port network传播延时propagation delay time显像管picture tube公式formula成正比proportional to成反比inversely proportional to/ vary indirectly with 带宽bandwidth幂power有理数rational exponent/number无理数irrational exponent/number磁场magnetic field天然的和人造的natural and artificial直线straight line曲线curved line镜头len二进制binary system十进制decimal system波形waveform摩擦friction探测detect雷达发射机radar transmitter防水water resistance光谱spectrum变压器transformer加减乘除plus, minus, multiplication, division比值，分数fraction电子望远镜electronic telescope光学望远镜optical telescope优越性superiority模-数转换器analog-to-digital converter (A/D’s)混合参数hybrid parameter功works动能kinetic energy万有引力gravitation, gravity, gravitational pull 位能potential energy气体膨胀gas expand核裂变nuclear fission接收机receiver放大器amplifier编码系统coding system天线antenna隧穿tunneling调制信号modulate signals数据处理data processing平移运动translational/ horizontal motion垂直运动perpendicular/ vertical motion进行试验conduct a test, do an experiment通讯卫星communication s satellite采样sampling样品specimen机械能mechanical energy电能electrical energy弹性elasticity导数derivative表示designate, denote幻灯机slide projector标准螺钉standard screw右旋right-handed串联in series并联in parallel动力学kinetic energy/ kinetics酶enzyme能力和局限性the capabilities and the limitations 中暑sunstroke画线plot a line图解graphical solution(火箭)推力thrust无限infinite,无数myriad无限大infinity, 无限小infinitesimal 有限limited, restricted, finite线圈coil皮带belt分析模型analytic model地磁场geomagnetic field金属导体metallic conductor微积分differential and integral calculus 指令instruction细胞质protoplasm细胞膜membrane周期波形periodic waveform循环系统circulatory system人体组织body tissue电流反馈current feedback容器vessel新奇事novelty绝缘体insulator漏电the leakage of current真空管vacuum tube倒数reciprocal浏览器browser磁感应magnetic induction装置apparatus, device, instrument电感器inductor误差error罗盘指针compass needle集电极collector晶体管transistor扩散率diffusivity变化情况variation导电率conductivity概率chance斜率slope滞后现象hysterresis实用性utility绝对值absolute极限值limit稍难词汇外部设备peripheral功耗dissipation流动flow曲线contour, curve谐振电路resonant circuit谐振频率resonant frequency电离层ionosphere矢量vector磁强度magnetic intensity热核能thermonuclear energy不能复制的unduplicated机械连接mechanical interconnection弧度radian氢hydrogen成立hold算法algorithm电动的electric红桃heart同时simultaneously只solely反相器inverter热效率thermal efficiency相应corresponding如下as follows常规仪器conventional instrument车床lathe曲线簇a specified set of curves平面plane载流导线current-carrying wire性质property极为重要的essential, paramount（至高无上的）不失真的undistorted无穷大infinity弹簧拉长A spring is stretched.类似于analogous to预设的predetermined曝光胶片expose the film惯性inertia谐振resonance百进制centesimal system计数器counter安培表ammeter增益函数gain coefficient力学原理the principles of mechanics质点/微粒particle锁向环phase-locked loop正弦信号sinusoidal signal非正弦信号nonsinusoidal signal收敛convergence令人讨厌的undesirable运动定律law of motion绕射diffraction回线loop二极管diode整流rectification欧姆表ohmmeter振幅amplitude里程计odometer分辨率resolution转换时间transition time重力势能gravitational potential energy可以忽略不计的negligible分离的数值discrete numerical value波纹ripple双滤波double filtering action用之不竭inexhaustible承受力withstand a force调谐放大器tuned amplifier经受考验stand the trial of高频振荡high-frequency oscillation阻尼振荡damped oscillation得到说明be accounted for by热流heat flow吸收水蒸气absorb water vapor干涉interference有弹性的elastic有出入，不一样deviation基极电流base current价电子valence electron脉冲输入pulse input电力反应堆power reactor阻抗impedance汗腺sweat gland对物体施加作用力exert force on an object联立方程组simultaneous equation线性方程组linear equation三角的、反三角的、指数的和对数函数trigonometric, inverse trigonometric, exponential and logarithmic functions对数logarithm易于......be susceptible / prone / liable / apt to单位时间in unit time二极管diode该方程无解There is no solution to the equation.铝aluminum积分路径the path of integration示波器oscilloscope内聚力cohesion最佳负载电荷optimum load resistance无线电回波radio echo标记法notation东西entity固态器件solid-state device分贝decibel(dB)微弱的非线性slight nonlinearities相对论力学relativistic mechanics戏称refer to it derisively圆函数circular function数列sequence不证自明self-evident平行四边形parallelogram四边形quadrangle五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon多项式polynomial对流热传递convective heat transfer感应电压induced voltage外加电压applied voltage阴极cathode阳极anticathode/ anode热阻thermal resistance冷却剂coolant蒸腾作用transpiration能力和局限性the capabilities and the limitations细胞质protoplasm滞后现象hysterresis。

IBDP数学课程简介

IBDP数学课程简介1.Introduction of IBDP Mathematics IBDP数学课程介绍Introduction:The nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably a combination of these, but there is no doubt that mathematical knowledge provides an important key to understanding the world in which we live. This prevalence of mathematics in our lives, with all its interdisciplinary connections, provides a clear and sufficient rationale for making this subject compulsory for students studying for the full diploma. 介绍：数学的本质可以用多种方式来归纳：例如，它可以被看作是一个被良好定义的知识体，是一个抽象的思想系统，或作为一个有用的工具。

对于很多人来说它可能是它们的组合，但毫无疑问的是数学知识是让我们了解所生活的世界的一个重要的关键。

数学在我们的生活中的普遍性，其所拥有的跨学科的联系，使得这一学课拥有充分理由成为每个学生的必修科目。

Aim:1.enjoy mathematics, and develop an appreciation of theelegance and power of mathematics2.develop an understanding of the principles and nature ofmathematics3.develop logical, critical and creative thinking, and patienceand persistence in problem-solving 目标：1.享受数学，发现并欣赏数学的优雅和力量。

IB数学单元知识

exponential functions
指数函数
derivatives of exponential functions
指数函数求导
logarithmetic functions
对数函数
derivatives of logarithmetic functions
对数函数求导
exponential growth and decay
最大最小值
applied maximum and minimum problems
最值问题的应用
trigonometric functions and their derivatives
trigonometric formulas
三角公式
trigonometric identities
三角恒等式
solving trgonometric equations
permutuations and combinations
排列组合
probability
概率
calculus（微积分-先修课程）
limits and rates of change
the limit of a function
函数的极限
one—sided limits
单边极限
velocity and other rates of change
单元知识
知识点
知识点说明
precalculus（微积分预备）
functions(函数）
functions and their graphs
函数和函数图像,微积分必备基础
linear and quadratic functions
一次函数与二次函数，国内初中内容

欧几里德数学竞赛简介EuclidContest滑铁卢大学UofWaterloo

欧几里德数学竞赛简介EuclidContest滑铁卢大学UofWaterloo滑铁卢大学(U of Waterloo)的欧几里德数学竞赛(Euclid Contest)---------加拿大“数学托福”的考试考试简介滑铁卢大学始建于1957年，在加拿大最权威的教育杂志Maclean`s（麦克林）的排名榜上，连续五年综合排名第一第二。

滑铁卢大学设有加拿大唯一一所数学学院，这也是北美乃至全世界最大的数学学院。

该学院拥有计算机科学、精算、生物信息、数学/会计、数学/工商管理等多种国际热门专业，每年接纳全世界105万名学生申请，却只有100名佼佼者能被录取。

面对众多的申请者，为了用客观、公平的标准来评估来评估学生的数学能力，滑铁卢大学最早1963年就采用了统一考试的办法。

最初的数学考试是由安大略省西南部的几个高中老师联合创办的，冲六十年代初年每年300人参加考试到今天，累计已经有21万名学生参加了这个考试。

根据滑铁卢大学的校方统计资料：21万名学生中有40%是来自安大略省的学生，20%是来自英属哥伦比亚省的学生，35%是来自加拿大其他省份的，还有5%是来自国际学生，包括美国、英国、中国等世界各国的学生。

2003年因为安大略省取消13年级，部分涉及微积分的试题不再使用，于是将迪卡尔（法国著名数学家）数学竞赛（Descartes Contest）更名为（欧几里德数学竞赛）。

现在，欧几里德数学竞赛的分数已经成为Waterloo数学学院各专业以及“软件工程”专业入学录取的重要指标，更成为学生申请该学院奖学金的重要考核标准。

欧几里德数学竞赛（Euclid Contest）主要是为高三年级（加拿大12年级）的高中学生提供的考试，考试内容主要包括：代数（函数、三角、排列、组合）、平面组合、解析几何等，他不仅仅看的是结果，更看重的是学生的解题思路和技巧。

考试的及格分数每年大概在40分左右。

因滑铁卢大学在数学领域的优良声誉及传统，以及欧几里德数学竞赛考察标准的严格性和专业性，该竞赛成绩在加拿大大学中已经得到广泛认可，被誉为类似加拿大“数学托福”的考试.考试形式考试时间为2.5小时，总共包括10道题，每题10分，总共100分。

指数_对数_幂函数必备知识点

指数_对数_幂函数必备知识点指数、对数和幂函数是数学中非常重要的概念和工具。

它们在各个领域中都有广泛的应用，包括科学、工程和经济等方面。

在这篇文章中，我们将详细介绍指数、对数和幂函数的必备知识点。

1. 指数函数（Exponential Functions）a.当a>1时，指数函数是递增函数，随着x的增加，函数值也增加；b.当0<a<1时，指数函数是递减函数，随着x的增加，函数值减小；c.当x=0时，f(x)=a^0=1；d.当x<0时，f(x)=a^x=1/a^(-x)。

指数函数在各个领域的应用非常广泛，比如在物理学中描述指数增长、衰变等现象，在经济学中描述复利现象等。

2. 对数函数（Logarithmic Functions）对数函数是指数函数的逆运算。

对数函数可以表示为f(x) =loga(x)，其中a为底数，x为正实数。

常见的对数函数有以10为底数的常用对数函数，即f(x) = log10(x) = lg(x)，以及以e为底数的自然对数函数，即f(x) = ln(x)。

对数函数具有以下特点：a.对数函数是递增函数，随着x的增加，函数值也增加；b. 当x=a时，f(a) = loga(a) = 1；c. 当x=1时，f(1) = loga(1) = 0；d.当a>1时，对数函数在定义域内的所有正实数上都有定义；e.当0<a<1时，对数函数只在定义域内的正实数中的一部分上有定义。

对数函数在数学和科学中有广泛的应用。

例如，对数函数可以用来解决指数方程、求解复利问题等。

3. 幂函数（Power Functions）幂函数是以x为底数，并以常数为指数的函数形式。

幂函数可以表示为f(x)=x^k，其中k为常数。

幂函数具有以下特点：a.当k>0时，幂函数是增函数，随着x的增加，函数值也增加；b.当k<0时，幂函数是减函数，随着x的增加，函数值减小；c.当k=0时，幂函数为常数函数，函数值始终为1幂函数在各个领域中都有广泛的应用。

函数英语作文模板

函数英语作文模板英文回答：Function Essay Template。

Introduction。

Define the term "function"State the purpose of the essay: to explore the various aspects of functions。

Body Paragraph 1: Types of Functions。

Discuss different types of functions, such as linear, quadratic, cubic, exponential, and logarithmic functions。

Provide examples of each type and explain their characteristics。

Body Paragraph 2: Properties of Functions。

Describe the key properties of functions, such as domain, range, increasing/decreasing intervals, concavity, and extrema。

Explain how to determine these properties from the function's graph or equation。

Body Paragraph 3: Applications of Functions。

Discuss the practical applications of functions in various fields, such as science, engineering, economics, and social sciences。

Provide specific examples of how functions are used to model real-world phenomena。

对数公式（Logarithmicformula）

对数公式（Logarithmic formula）Exponential function and logarithmic functionKey points and difficulties:Emphasis: the concepts, images, and properties of exponential functions and logarithmic functions.Difficulties: the relation between exponential function and logarithmic function, the application of the property, and the idea of logical division, and discuss the function in two different situations.1, exponential function:Definition: functions are called exponential functions.The domain of R, the base is constant, the index is the independent variable.Why is the a required in the function required?.Because if the time, and then, the function value does not exist.When the function value does not exist.The value of the function is always 1, but the inverse function does not exist because of all the X, because the function is required.1. Understanding of the images of three exponential functions.Image features and functional properties:Property of image characteristic function(1) images are located in the X axis; (1) there are x from any real value,; (2) image after a point (0, 1); (2) whether a take any positive, when; (3) the ordinate in the first quadrant are greater than 1, the ordinate in the second quadrant are less than 1, the image on the contrary;(3) at that time,Then,(4) the image gradually rises from left to right, and the image decreases gradually.(4) at that time, it was an increasing function,At that time, the function was reduced.Further understanding of images (by comparison of three functions):The images of all exponential functions intersect at points (0, 1), such as and intersect, at that time, the image is above the image, and, when exactly the opposite, there is.And the image about y axis symmetry.By three, and the image function, you can draw an arbitrary function () diagram, such as the image, and is located in the middle of certain of the two images, and thus, by symmetry on the Y axis, to sketch, through the image of finite function understanding infinite function image.2, logarithms:Definition: if so, the number of B is called the logarithm base a, denoted by (log a, N is a natural, is logarithmic.)As a result, the N must be greater than 0.The logarithm does not exist when the N is zero negative.(1) logarithmic and exponential interaction.Since logarithms are new, they often transform unfamiliar logarithms into exponentially solving problems, such as:seekAnalysis: for beginners, the above problem is generally helpless, if it is written, and then rewritten to index type is easier to handle.Solution: set upComment: it can solve the problem from logarithmic to exponential, but it can solve the problem by index to logarithm,so it must be different. Such as in, into a logarithmic formula that is.(2) logarithmic identities:fromTo substitute (2) for (1)The use of logarithmic identities to the attention of the characteristics of this type, can not be used indiscriminately, especially when the power conversion must pay attention to the base base and the logarithm of the same.Calculation：Solution: primitive.(3) the nature of logarithms:There is no logarithm for the negative and zero;The logarithm of 1 is zero;The logarithmic base is equal to 1.(4) the arithmetic of logarithms:1.234.3, logarithmic function:Definition: the inverse function of an exponential function is called a logarithmic function.1 pairs of three logarithmic functionsUnderstanding of images.Image features and functional properties:Property of image characteristic function(1) images are located on the right side of the Y axis; (1) define domains: R+, values, or: R;(2) the image is over (1, 0); (2),. That is, (3) at that time, the image was above the X axis, when the image was below the X axis, which was just the opposite;(3) at that time, if, then, if;At that time, if, then, if, then, (4) from left to right, the image is rising, while from left to right the image is down.(4) is an increasing function;Is a decreasing function. A further understanding of images (by comparison of three functions, images, or correlations):(1) images of all logarithmic functions are point (1,0), but at the point (1,0) curve is a cross, at that time, in the image above the image; and, in the image below, it is:;.(2) an image and an image about X axis symmetry.(3) by three, and the image function, can make a sketch of an arbitrary logarithmic function, such as image, it must be located in the middle and two images, and the cross point (1,0), and in the above, and at the bottom, when, on the contrary, it is symmetry. Can know the sketch map.Thus, the image of an infinite function is further recognized by the image of three functions in the textbook.4, the logarithm of the formula:The formula can be obtained by:Introduce some useful conclusions by the formula:(1)(2)(3)(4)5. Exponential equation and logarithmic equation *Definition: an equation containing an unknown number in an exponential equation.The equation that contains an unknown number behind a logarithmic symbol is called a logarithmic equation.Because exponential operation and logarithmic operation are not general algebraic operations, exponential equation and logarithmic equation are not algebraic equations, but belong to transcendental equations.The questions and solutions of exponential equation:Name question type solutionBasic typeThe same base typeDifferent base typeSubstitutional typeTake the logarithm based on aTake the logarithm based on aTake the logarithm of the same base asThe question type and solution of logarithmic equation of algebraic equation converted from binary order:Name question type solutionBasic questionsLog into the same base type index typeChange to (must examine root) to need to replace typeThe transformation order is transformed into algebraic equation。