Barron's AP Calculus BOOK_01
APCalculusABreviewAP微积分复习提纲PDF.pdf
APCalculusABreviewAP微积分复习提纲PDF.pdfAP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIf f is defined on an open interval containing c, and if the limitexists, then the line passing through (c, f(c)) with slope m is the tangentline to the graph of f at the point (c, f(c)).Definition of the Derivative of a FunctionThe Derivative of f at x is given byprovided the limit exists. For all x for which this limit exists, f’is afunction of x.*The Power Rule*The Product Rule***The Chain RuleImplicit Differentiation (take the derivative on both sides; derivativeof y is y*y’)Chapter 3Applications of Differentiation*Extrema and the first derivative test (minimum: ? → + , maximum: +→ ?, + & ? are the sign of f’(x) )*Definition of a Critical NumberLet f be defined at c. If f’(c) = 0 OR IF F IS NOT DIFFERENTIABLEAT C, then c is a critical number of f.*Rolle’s TheoremIf f is differentiable on the open interval (a, b) and f (a) = f (b), then thereis at least one number c in (a, b) such that f’(c) = 0.*The Mean Value TheoremIf f is continuous on the closed interval [a, b] and differentiable on theopen interval (a, b), then there exists a number c in (a, b) such that f’(c) = .*Increasing and decreasing interval of functions (take the first derivative)*Concavity (on the interval which f’’ > 0, concave up)*Second Derivative TestLet f be a function such that f’(c) = 0 and the second derivative of f existson an open interval containing c.1.If f’’(c) > 0, then f(c) is a minimum2.If f’’(c) < 0, then f(c) is a maximum*Points of Inflection (take second derivative and set it equal to 0, solve theequation to get x and plug x value in original function)*Asymptotes (horizontal and vertical)*Limits at Infinity*Curve Sketching (take first and second derivative, make sure all thecharacteristics of a function are clear)Optimization Problems*Newton’s Method (used to approximate the zeros of a function, which istedious and stupid, DO NOT HA VE TO KNOW IF U DO NOT WANTTO SCORE 5)Chapter 4 & 5Integration*Be able to solve a differential equation*Basic Integration Rules1)2)3)4)*Integral of a function is the area under the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculate the area for each sub-interval and summation is the integral).*Definite integral*The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a, b] and F is an anti-derivative of f on the interval [a, b], then.*Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a, b], then the average value off on the interval is.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then, forevery x in the interval,.*Integration by Substitution*Integration of Even and Odd Functions1) If f is an even function, then.2) If f is an odd function, then.*The Trapezoidal RuleLet f be continuous on [a, b]. The trapezoidal Rule for approximating is given byMoreover, a n →∞, the right-hand side approaches.*Simpson’s Rule (n is even)Let f be continuous on [a, b]. Simpson’s Rule for approximating isMoreover, as n→∞, the right-hand side approaches*Inverse functions(y=f(x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f’(g(x))≠0. Moreover,, f’(g(x))≠0.*The Derivative of the Natural Exponential FunctionLet u be a differentiable function of x.1. 2..*Integration Rules for Exponential FunctionsLet u be a differentiable function of x..Derivatives for Bases other than eLet a be a positive real number (a ≠1) and let u be adifferentiable function of x.1. 2.*Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x.*Definition of the Hyperbolic Functions。
AP Calculus 课程
AP Calculus 课程AP Calculus课程考试设置了两个考试：Calculus AB和 Calculus BC。
后者比起前者只是增加了部分内容。
该AP课程无论从所涵盖内容和对学生的要求，都和在大学里所修的Calculus课程没有差别。
Calculus 是美国大学理工科学生必修的一门数学课程，中文名为微积分，在中国大陆的大学里也是一门课程，被称作高等数学。
微积分学是微分学和积分学的总称。
它是一种数学思想，‘无限细分’就是微分，‘无限求和’ 就是积分。
微积分主要内容包括极限、微分学、积分学及其应用。
微分学包括求导数的运算，是一套关于变化率的理论。
它使得函数、速度、加速度和曲线的斜率等均可用一套通用的符号进行讨论。
积分学，包括求积分的运算，为定义和计算面积、体积等提供一套通用的方法。
微积分是与实际应用联系着发展起来的，它在天文学、力学、化学、生物学、工程学、经济学等自然科学、社会科学及应用科学等多个分支中，有越来越广泛的应用。
特别是计算机的发明更有助于这些应用的不断发展。
AP Calculus 考试Calculus AB和 Calculus BC课程考试各需要3小时15分钟。
每门考试都包含105分钟的选择题部分和90分钟的自由回答题部分。
两部分所占分值相等，部分题目需要用计算器才能完成。
博美前程AP Calculus课程的特点博美前程在北美有着多年的AP Calculus教学经验。
所有老师均由英美及国内知名高校的硕士、博士研究生担纲教学，并通过博美严格的系统培训。
博美的目标是使得学生不仅能够在考试中取得好成绩，也让他们真正学好学透微积分课程，为他们在今后大学学习中打下扎实的基础。
AP Statistics课程AP统计学课程考试面向高中学生。
通过考试的学生等同于在无微积分基础的条件下，掌握了统计学的入门课程，相当于大学一学期的学习水平。
AP Statistics课程包含四个主题：数据探究，抽样和试验，预期模式和统计推断。
AP考试模拟试题与答案1-微积分BC- AP Calculus-BC
(A) III only (B) I and II only
(C) II and III only (D) I and III only (E) I, II, and III
y f
a
0
x b
Figure 1T-3
7.
∞
1
=
n = 1 (2n − 1)(2n + 1)
(C) e 2
x
a
0
b
(A)
y
Figure 1T-1 A possible graph of f is (see Figure 1T-2):
(B)
y
(C)
y
a
b
x
a0
b
xa
0
bx
(D)
y
(E)
y
a0
b
x
a
0
bx
Figure 1T-2
GO ON TO THE NEXT PAGE
374 STEP 5. Build Your Test-Taking Confidence
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AP Calculus BC Practice Exam 1 373
Section I—Part A
Number of Questions 28
Time 55 Minutes
Use of Calculator No
Directions:
ap考试模拟试题与答案1微积分bcapcalculusbc
AP Calculus BC Practice Exam 1 371
AP Calculus BC Practice Exam 1
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,
calculus我的calculus笔记,按照国际标准排版
1.2
Mathematical Models: A Catalog of Essential Functions
1. A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon. 2. If there is no physical law or principle to help us formulate a model, we construct an empirical model. 3. A function P is called a polynomial if P pxq an xn an¡1 xn¡1 ¤ ¤ ¤ a2 x2 a1 x a0 when n N and the numbers a0 , a1 , a2 , . . . , an are constants called coefficients of the polynomial. The domain of any polynomial is R. If the leading coefficient an $ 0, then the degree of the polynomial is n. A polynomial of degree 1 is of the form P pxq mx b and is a linear function. A polynomial of degree 2 is of the form P pxq ax2 bx c and is called a quadratic function.
1
CHAPTER 1
APCalculusABChapter4,Section1-CorsicanaISD
_____________________________.
• Integrals are often called ________________.
Derivatives vs. Integrals
• How are the three given functions related?
(Use -32 feet per second as the acceleration due to gravity.)
Ch. 4.1 Homework
• Pg 255 – 257, #’s: 1, 7, 9, 13, 15, 23, 29, 35, 39, 45, 55, 61, 69
• 13 total problems
•
(3 4 − 5 2 + )
Rewriting Before Integrating
•
+1
Rewrite Before Integrating
•
sin
s2 x
Initial Conditions and Particular
Solutions
• The type of antiderivatives you have been learning about are
1
3
2 sin
Rewrite
Integrate
Simplify
Integrating Polynomial
Functions
•
Integrating Polynomial
Functions
•
2017 ap calculusab 微积分
2017 ap calculusab 微积分英文版2017 AP Calculus AB: A Journey Through MicrocalculusAs the sun rose over the horizon, students across the globe began their journey into the world of mathematics with the 2017 AP Calculus AB exam. This exam, known for its depth and breadth, tests the student's understanding of the fundamental concepts of calculus.The exam began with a gentle reminder of the basic derivative rules, followed by questions that required students to apply these rules to real-world scenarios. One such question dealt with the optimization of a profit function, testing the student's ability to identify the maximum or minimum value of a function. This question highlighted the practical applications of calculus in real-life situations.As the exam progressed, the questions became more complex, delving into the realm of integration. Students werechallenged to evaluate integrals using various techniques, such as substitution and integration by parts. One notable question dealt with the concept of areas between curves, requiring students to apply their knowledge of integration to find the enclosed area.The exam also included questions on sequences and series, testing the student's understanding of convergence and divergence. Questions on infinite series were particularly challenging, as they required students to analyze the behavior of the series as it approached infinity.Towards the end of the exam, students were presented with a challenging question on differential equations. This question tested their ability to understand and manipulate differential equations, ultimately finding a solution that satisfied the given conditions.The 2017 AP Calculus AB exam was not just a test of mathematical knowledge; it was a testament to the students' dedication, perseverance, and understanding of the beauty andpower of calculus. As the sun set, students closed their books, satisfied that they had done their best, and hoped that their efforts would be rewarded with a smile on the faces of their teachers and parents.中文版2017 AP微积分AB：微积分之旅随着太阳从地平线上升起，全球的学生们开始了他们的数学之旅，参加了2017年的AP微积分AB考试。
AP考试模拟试题与答案1-微积分BC- AP Calculus-BC
1
14
(A) (B) 1 (C)
(D) 4 (E) 5
2
16
15. Which of the following is an equation of the line tangent to the curve with parametric equations x = 3t2 − 2, y = 2t3 + 2 at the point when t = 1?
−
d 3)
x
=
n
5x
(A) lim n→0
−3
(x
+
2)(x
−
dx 3)
−2
5x
(B) lim n→− 3 + n
(x
+
2)(x
−
dx 3)
n
5x
(C) lim n→− 2−
− 3 (x + 2)(x − 3) d x
n
5x
(D) lim n→− 3
ห้องสมุดไป่ตู้
−3
(x
+
2)(x
−
dx 3)
−2
5x
(E) lim n→0 n
III. f < 0 on (0, b)
(A) III only (B) I and II only
(C) II and III only (D) I and III only (E) I, II, and III
y f
a
0
x b
Figure 1T-3
7.
∞
1
=
n = 1 (2n − 1)(2n + 1)
Calculus_BC_Syllabus_1[1] AP微积分教学大纲
![Calculus_BC_Syllabus_1[1] AP微积分教学大纲](https://img.taocdn.com/s3/m/bda3a63e0912a216147929c2.png)
oo Substitution, integration by parts, trigonometric substitution, partial fractions
• Separable differential equations
• Euler’s Method
• Taylor’s series/Maclaurin series • Lagrange form of the remainder • Tests for convergence/divergence:
oo nth term test oo Direct Comparison oo Ratio Test oo Integral Test oo Limit Comparison Test oo Alternating Series Test (Leibniz’s Theorem)
The chapter numbers follow the textbook. Note that we work on Chapter 10 before Chapter 9.
Chapter 1: Prerequisites for Calculus (7 days)
• Elementary functions:
Chapter 8: L’Hôpital’s Rule, Improper Integrals,
Partial Fractions (13 days)
• Indeterminate forms
⎛0 ⎜
,
∞
,∞
−
⎞ ∞ , 1∞ , 0 0 , ∞ 0 ⎟
and L’HÔpital’s Rule
AP学习备考必看书单推荐
AP学习备考必看书单推荐为了帮助大家学会AP考试技巧，熟悉AP备考必看书单，AP频道为大家带来2018年AP学习备考必看书单推荐一文，希望对大家AP备考有所帮助。
微积分by.Yvonne教材推荐：1 James Stewart Calculus 7th editionStewart Calculus 的最新版本，难度略高于考试。
本书将微积分考试中所有公式进行了推导和证明，每章均配有大量例题和练习题。
考生在使用此书的同时，可以熟悉、掌握微积分公式的用法，打下坚实基础。
2 Calculus: AP Edition 9th Edition by RonLarson, Bruce H. Edwards这本书的编排很好，便于学生理解，在内容方面它涉及到了AP微积分在实际生活当中的运用，使知识不再抽象。
本书还配有一个学习网站，学生可以在该网站查看练习题解析。
教辅推荐：* 巴朗Barron’s♪统计学By. Yvonne & 李抒彦(founder of Thrive 北美人物志)教材推荐：1 The Practice of Statistics by Daren S.Starnes本书的结构适合递进式学习，尤其在Hypothesis一章加入了大量事实案例。
关于AP 考试的要点解释得非常到位。
教辅推荐：*三立AP统计学;* 普林斯顿统计学知识点讲的详细，但讲计算器用法那章不太好懂，但课后习题太简单。
♪化学BY. Yvonne教材推荐：本书的布局方式十分简洁，可以顺利地从初中知识所学直接过渡到AP Chemistry。
本书逻辑线清晰，讲解明了，并附有很多习题，着实是一本很好的教材。
优点：包含了所有AP Chem的考点，并且每一章节的练习题十分充足。
缺点：练习题种类太多，而且有的问题问的有些不知所云。
教辅推荐：* 巴朗Barron’s;* 普利斯顿Princeton Review;* 美国化学竞赛选择题♪生物By. 安世文教材推荐：* Campell BiologyUnsurpassed leader in introductory biology.非常棒的一本入门书籍，囊括了所有基础生物学内容，在国内只有陈阅增先生的普通生物学能与之媲美。
AP calculus formula sheet
final position - initial position total time
e x = 1+ x + cos( x) = 1 −
x 2 x3 + + 2 3! x2 x 4 + − 2 4! x3 x5 + − 3! 5!
+ Polar Curves For a polar curve r(θ), the area inside a "leaf" is
∫ ln( x ) dx = x ln( x ) − x + C
+ l'Hôpital's Rule
N
where θ1 and θ2 are the "first" two times that r = 0. The slope of r(θ) at a given θ is
+ Alternating Series Error Bound If S N =
(
)
∫ f ( x ) dx ≈
a
b
f ( x0 ) + f ( x1 ) Δx1 + 2 f ( xn − 1 ) + f ( xn ) Δxn 2
( (
) )
…+
d 1 sec−1 ( x ) = dx x x2 − 1 d −1 csc−1 ( x ) = dx x x2 −1
( (
) )
Solids of Revolution and friends Volume
(n +1)
+ Euler's Method
dy = f (x , y ) and that the dx solution passes through ( x 0 , y0 ) ,
AP 微积分BC 选择题样卷一
AP Calculus Practice ExamBC Version - Section I - Part ACalculators ARE NOT Permitted On This Portion Of The Exam28 Questions - 55 Minutes1) GivenFind dy/dx.a)b)c)d)e)2) Give the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1 and x = e, about the y-axis.a)b)c)d)e)3) The graph of the derivative of f is shown below.Find the area bounded between the graph of f and the x-axis over the interval [-2,1], given that f(0) = 1.a)b)c)d)e)4) Determine dy/dt, given thatanda)b)c)d)e)5) The functionis invertible. Give the slope of the normal line to the graph of f -1 at x = 3.a)b)c)d)e)6) Determinea)b)c)d)e)7) Give the polar representation for the circle of radius 2 centered at ( 0 , 2 ).a)b)c)d)e)8) Determinea)b)c)d)e)9) Determinea)b)c)d)e)10) Give the radius of convergence for the seriesa)b)c)d)e)11) Determinea)b)c)d)e)12) The position of a particle moving along the x-axis at time t is given byAt which of the following values of t will the particle change direction I) t = 1/8II) t = 1/6III) t = 1IV) t = 2a) I, II and IIIb) I and IIc) I, III and IVd) II, III and IVe) III and IV13) Determinea)b)c)d)e)14) Determine the y-intercept of the tangent line to the curveat x = 4.a)b)c)d)e)15) The function f is graphed below.Give the number of values of c that satisfy the conclusion of the Mean Value Theorem for derivatives on the interval [2,5].a)b)c)d)e)16) Give the average value of the functionon the interval [1,3].a)b)c)d)e)17) A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 20 square feet. Give the rate of change of the width (in ft/sec) when the height is 5 feet, if the height is decreasing at that moment at the rate of 1/2 ft/sec.a)b)c)d)e)18) The graph of the derivative of f is shown below.Give the number of values of x in the interval [-3,3] where the graph of f has inflection.a)b)c)d)e)19) A rectangle has its base on the x-axis and its vertices on the positive portion of the parabolaWhat is the maximum possible area of this rectanglea)b)c)d)e)20) Computea)b)c)d)e)21) Determinea)b)c)d)e)22) Determinea)b)c)d)e)23) Give the exact value ofa)b)c)d)e)24) Determinea)b)c)d)e)25) Give the derivative ofa)b)c)d)e)26) Give the first 3 nonzero terms in the Taylor series expansion about x = 0 for the functiona)b)c)d)e)27) Determinea)b)c)d)e)28) Which of the following series converge(s)a) B onlyb) A, B and Cc) B and Cd) A and Be) A and C1) d)2) e)3) b)4) b)5) e)6) a)7) c)8) c)9) b)10) d)11) c)12) c)13) c)14) e)15) a)16) d)17) a)18) b)19) a)20) d)21) b)22) b)23) a)24) c)25) d)26) b)27) b)28) c)。
AP Calculus AB review AP微积分复习提纲PDF.pdf
AP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIf f is defined on an open interval containing c, and if the limitexists, then the line passing through (c, f(c)) with slope m is the tangentline to the graph of f at the point (c, f(c)).Definition of the Derivative of a FunctionThe Derivative of f at x is given byprovided the limit exists. For all x for which this limit exists, f’is afunction of x.*The Power Rule*The Product Rule***The Chain Rule☺Implicit Differentiation (take the derivative on both sides; derivativeof y is y*y’)Chapter 3Applications of Differentiation*Extrema and the first derivative test (minimum: − → + , maximum: +→ −, + & − are the sign of f’(x) )*Definition of a Critical NumberLet f be defined at c. If f’(c) = 0 OR IF F IS NOT DIFFERENTIABLEAT C, then c is a critical number of f.*Rolle’s TheoremIf f is differentiable on the open interval (a, b) and f (a) = f (b), then thereis at least one number c in (a, b) such that f’(c) = 0.*The Mean Value TheoremIf f is continuous on the closed interval [a, b] and differentiable on theopen interval (a, b), then there exists a number c in (a, b) such that f’(c) = .*Increasing and decreasing interval of functions (take the first derivative)*Concavity (on the interval which f’’ > 0, concave up)*Second Derivative TestLet f be a function such that f’(c) = 0 and the second derivative of f existson an open interval containing c.1.If f’’(c) > 0, then f(c) is a minimum2.If f’’(c) < 0, then f(c) is a maximum*Points of Inflection (take second derivative and set it equal to 0, solve theequation to get x and plug x value in original function)*Asymptotes (horizontal and vertical)*Limits at Infinity*Curve Sketching (take first and second derivative, make sure all thecharacteristics of a function are clear)♫ Optimization Problems*Newton’s Method (used to approximate the zeros of a function, which istedious and stupid, DO NOT HA VE TO KNOW IF U DO NOT WANTTO SCORE 5)Chapter 4 & 5Integration*Be able to solve a differential equation*Basic Integration Rules1)2)3)4)*Integral of a function is the area under the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculate the area for each sub-interval and summation is the integral).*Definite integral*The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a, b] and F is an anti-derivative of f on the interval [a, b], then.*Definition of the Average Value of a Function on an IntervalIf f is integrable on the closed interval [a, b], then the average value off on the interval is.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then, for every x in the interval,.*Integration by Substitution*Integration of Even and Odd Functions1) If f is an even function, then.2) If f is an odd function, then.*The Trapezoidal RuleLet f be continuous on [a, b]. The trapezoidal Rule for approximating is given byMoreover, a n →∞, the right-hand side approaches.*Simpson’s Rule (n is even)Let f be continuous on [a, b]. Simpson’s Rule for approximating isMoreover, as n→∞, the right-hand side approaches*Inverse functions(y=f(x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f’(g(x))≠0. Moreover,, f’(g(x))≠0.*The Derivative of the Natural Exponential FunctionLet u be a differentiable function of x.1. 2..*Integration Rules for Exponential FunctionsLet u be a differentiable function of x..♠Derivatives for Bases other than eLet a be a positive real number (a ≠1) and let u be a differentiable function of x.1. 2.♠♠*Derivatives of Inverse Trigonometric FunctionsLet u be a differentiable function of x.*Definition of the Hyperbolic Functions。
AP微积分-APCalculus公式大全-217
AP微积分-APCalculus公式⼤全-217 AP Calculus BC1. Important limits()001111011sin sin lim1, lim 1lim 1lim 10, ()lim lim , x x xt x t m m m m m m n n x x x ax ax bx be t e x m n P x a x a x a x a a m n b x b →→→∞→---→∞→∞==+=?+=<++++===+++??x x sec )'(tan = x csc (cot)'-= x x x tan sec )'(sec = x x x cot csc )'(csc -=(6) 211)'(arcsin xx -=211)'(arccos xx --=211)'(arctan x x +=211)'cot (x x ar +-= 11)'sec (2-=x x x arc 11)'csc (2--=x x x arc2. Rules（1）If f (x )，g (x ) are differential ，a. )()())()((x g x f x g x f '±'='±；b. )()()()())()((x g x f x g x f x g x f '+'='，especially ，)())((x f C x Cf '='（C is a constant ）；c. )0)(( ,)()()()()())()((2≠'-'='x g x g x g x f x g x f x g x f ，especially ，21()()()()g x g x g x ''=-。