SAT数学课程第13课(Radical s and Triangles)
SAT数学词汇
知识分块版1.Arithmetic (算数部分) Number theory (数论)natural number自然数positive number正数negative number负数odd integer, odd number奇数even integer, even number偶数integer, whole number整数positive whole number正整数negative whole number负整数consecutive whole number连续整数real number, rational number实数,有理数irrational(number)无理数composite number合数e.g.4,6,8,9,10,12,14,15……prime number质数e.g.2,3,5,7,11,13,15……reciprocal倒数（greatest）common divisor(最大)公约数（greatest）common factor (最大)公因子multiple倍数(least)common multiple(最小)公倍数(prime)factor(质)因子ordinary scale, decimal scale十进制nonnegative非负的Fraction and Decimal(分数和小数) tens place十位hundreds place 百位thousands place 千位units, ones个位tenths place 十分位hundredths place 百分位common ratio公比proper fraction真分数improper fraction假分数mixed number带分数vulgar fraction，common fraction普通分数simple fraction简分数complex fraction繁分数numerator分子denominator分母(least)common denominator(最小)公分母quarter四分之一Operation (基本数学运算)add, plus加subtract减difference差multiply, times乘divide除product积divisible可被整除的divided evenly被整除dividend被除数divisor因子，除数quotient商remainder余数factorial阶乘power乘方radical sign, root sign根号round to四舍五入to the nearest四舍五入Sequence (数列)arithmetic progression(sequence)等差数列geometric progression(sequence)等比数列Set (集合)union并集intersection 交集complementary 补集proper subset真子集solution set解集Others (其他)approximate近似(anti)clockwise(逆)顺时针方向cardinal基数ordinal序数distinct不同的estimation估计，近似parentheses括号proportion比例table表格trigonometric function三角函数unit单位,位2.Algebra (代数部分)algebraic term代数项like terms, similar terms同类项numerical coefficient数字系数literal coefficient字母系数inequality不等式triangle inequality三角不等式range值域original equation原方程equivalent equation同解方程/等价方程linear equation线性方程(e.g. 5x+6=22) exponent指数，幂base底边，底数(e.g.2的5次方，2就是底数) base乘幂的底数,底边cube立方数，立方体square root平方根cube root立方根common logarithm常用对数digit数字constant常数variable变量inverse function反函数linear一次的，线性的factorization因式分解absolute value绝对值，e.g.｜-32｜=32 round off四舍五入directly proportion to 正比inversely proportion to 反比3.Geometry(几何部分)Angles (角)alternate angle内错角corresponding angle同位角vertical angle对顶角central angle圆心角interior angle内角exterior angle外角supplementary angles补角complementary angle余角adjacent angle邻角acute angle锐角obtuse angle钝角right angle直角round angle周角straight angle平角included angle夹角Triangles (三角形)equilateral triangle等边三角形scalene triangle不等边三角形isosceles triangle等腰三角形right triangle直角三角形oblique斜三角形inscribed triangle内接三角形Other Polygons (其他多边形) semicircle半圆concentric circles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形octagon八边形nonagon九边形decagon十边形polygon多边形parallelogram平行四边形equilateral等边形plane平面square正方形，平方rectangle长方形regular polygon正多边形rhombus菱形trapezoid梯形Other Plane Graphs(平面图形) arc弧line, straight line直线line segment线段parallel lines平行线segment of a circle弧形Solids (立体图形)cube立方体，立方数rectangular solid长方体regular solid/regular polyhedron正多面体circular cylinder圆柱体cone圆锥sphere球体solid立体的Coordinate system (坐标系) coordinate system坐标系rectangular coordinate直角坐标系origin原点abscissa横坐标ordinate纵坐标number line数轴quadrant象限slope斜率intercept截距Othersplane geometry平面几何trigonometry三角学bisect平分circumscribe外切inscribe内切intersect相交perpendicular垂直Pythagorean theorem勾股定理congruent全等的multilateral多边的altitude高depth深度side边长circumference, perimeter周长radian弧度surface area 表面积volume体积arm直角三角形的股cross section横截面center of a circle圆心chord弦radius半径angle bisector角平分线diagonal对角线diameter直径edge棱face of a solid立体的面hypotenuse斜边included side夹边leg三角形的直角边median of a triangle三角形的中线opposite直角三角形中的对边midpoint中点endpoint端点vertex (复数形式vertices)顶点tangent切线的transversal截线4.Probability and Statistics(概率和统计部分)mode众数median中位数mean 平均数arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数permutation排列combination组合circle graph( pie graph) 饼图line graph 折线图bar graph 柱状图pictograph 象形图词典版 A(anti)clockwise(逆)顺时针方向 abscissa 横坐标absolute value 绝对值，e.g.｜-32｜=32 acute angle 锐角 add, plus 加adjacent angle 邻角 algebraic term 代数项 alternate angle 内错角 altitude 高angle bisector 角平分线 approximate 近似 arc 弧arithmetic mean 算术平均值arithmetic progression(sequence)等差数列 arm 直角三角形的股Bbar graph 柱状图base 乘幂的底数,底边base 底边，底数(e.g.2的5次方，2就是底数) bisect 平分Ccardinal 基数center of a circle 圆心 central angle 圆心角 chord 弦circle graph( pie graph) 饼图circular cylinder 圆柱体circumference, perimeter 周长 circumscribe 外切（greatest ）common divisor(最大)公约数（greatest ）common factor (最大)公因子 (least)common denominator(最小)公分母 (least)common multiple(最小)公倍数 complementary 补集complex fraction 繁分数composite number 合数e.g.4,6,8,9,10,12,14,15…… concentric circles 同心圆 cone 圆锥congruent 全等的consecutive number 连续整数constant 常数coordinate system 坐标系 corresponding angle 同位角 cross section 横截面 cube root 立方根cube 立方数，立方体 cube 立方体，立方数Ddecagon 十边形 denominator 分母 depth 深度diagonal 对角线 diameter 直径 difference 差 digit 数字directly proportion to 正比 distinct 不同的divided evenly 被整除 dividend 被除数 divide 除divisible 可被整除的 divisor 因子，除数Eedge 棱endpoint 端点equilateral triangle 等边三角形 equilateral 等边形equivalent equation 同解方程/等价方程 estimation 估计，近似even integer, even number 偶数 exponent 指数，幂 exterior angle 外角F face of a solid 立体的面 (prime)factor(质)因子 factorization 因式分解G geometric mean 几何平均数 geometric progression(sequence)等比数列H heptagon 七边形hexagon六边形hundreds place 百位hundredths place 百分位hypotenuse斜边Iimproper fraction假分数included angle夹角included side夹边inequality不等式inscribed triangle内接三角形inscribe内切integer, whole number整数intercept截距interior angle内角intersection 交集intersect相交inverse function反函数inversely proportion to 正比irrational(number)无理数isosceles triangle等腰三角形Lleg三角形的直角边like terms, similar terms同类项line graph 折线图line segment线段line, straight line直线linear equation线性方程(e.g. 5x+6=22) linear一次的，线性的literal coefficient字母系数Mmean 平均数median of a triangle三角形的中线median中位数midpoint中点mixed number带分数mode众数multilateral多边的multiple倍数multiply, times乘Nnatural number自然数negative number负数negative whole number负整数nonagon九边形nonnegative非负的number line数轴numerator分子numerical coefficient数字系数Ooblique斜三角形obtuse angle钝角octagon八边形odd integer, odd number奇数opposite直角三角形中的对边ordinal序数ordinary scale, decimal scale十进制ordinate纵坐标original equation原方程origin原点Pparallel lines平行线parallelogram平行四边形parentheses括号pentagon五边形permutation排列perpendicular垂直pictograph 象形图plane geometry平面几何plane平面polygon多边形positive number正数positive whole number正整数power乘方prime number质数e.g.2,3,5,7,11,13,15……product积proper fraction真分数proper subset真子集proportion比例Pythagorean theorem勾股定理Qquadrant象限quadrilateral四边形quarter四分之一quotient商Rradian弧度radical sign, root sign根号radius半径range值域real number, rational number实数,有理数reciprocal倒数rectangle长方形rectangular coordinate直角坐标系rectangular solid长方体regular polygon正多边形regular solid/regular polyhedron正多面体remainder余数rhombus菱形right angle直角right triangle直角三角形round angle周角round off四舍五入round to四舍五入Sscalene triangle不等边三角形segment of a circle弧形semicircle半圆side边长simple fraction简分数slope斜率solid立体的solution set解集sphere球体square root平方根square正方形，平方straight angle平角subtract减supplementary angles补角surface area 表面积Ttable表格tangent切线的tens place十位tenths place 十分位thousands place 千位to the nearest四舍五入transversal截线trapezoid梯形triangle inequality三角不等式trigonometric function三角函数trigonometry三角学Uunion并集units, ones个位unit单位,位Vvariable变量vertex (复数形式vertices)顶点vertical angle对顶角volume体积vulgar fraction，common fraction普通分数Wweighted average加权平均值注：文中部分内容来自网络。
unit13九年级知识点
unit13九年级知识点Unit 13: 九年级知识点Unit 13 is an important unit in the ninth-grade curriculum. In this unit, students will learn various key concepts and knowledge that are essential for their academic growth. This article aims to provide a comprehensive overview of the main topics covered in this unit.1. Solving Equations:One of the fundamental skills students will develop in Unit 13 is solving equations. They will learn different methods such as the balance method and using inverse operations to find the value of an unknown variable. Solving equations is a crucial mathematical skill as it allows us to find unknown values and represent relationships between variables.2. Inequalities:Building upon the concept of equations, students will also learn about inequalities. Inequalities express relationships between variables using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding inequalities isessential for solving real-world mathematical problems, such as determining the number of solutions that satisfy certain conditions.3. Graphing Linear Equations:Another important topic covered in Unit 13 is graphing linear equations. Students will learn how to plot points on a coordinate grid and connect them to form a line. Graphing linear equations helps visualize the relationship between two variables and provides a graphical representation of the equation. This skill is crucial in various fields, including science, engineering, and economics.4. Proportional Relationships:Unit 13 also introduces students to the concept of proportional relationships. Proportional relationships exist when two quantities vary in a consistent manner. Students will learn how to identify proportional relationships, represent them using equations or tables, and solve problems involving proportional reasoning. This knowledge is valuable in various real-life scenarios, especially when dealing with rates, ratios, and percentages.5. Similarity and Congruence:Furthermore, students will delve into the realm of geometry in Unit 13 by exploring similarity and congruence. Similar figures have the same shape but differ in size, while congruent figures have both the same shape and size. Students will learn how to identify and apply properties of similar and congruent triangles. These concepts are crucial in fields such as architecture, design, and engineering.6. Pythagorean Theorem:Unit 13 also covers the Pythagorean Theorem. This theorem relates the lengths of the sides of a right triangle and is considered one of the most important theorems in mathematics. Students will learn how to apply the Pythagorean Theorem to solve problems involving right triangles and calculate unknown side lengths. This theorem has significant applications in various fields, including architecture, navigation, and physics.In conclusion, Unit 13 encompasses a range of important mathematical concepts and skills. From solving equations to graphing linear equations, from understanding inequalities to exploring similarity and congruence, and from proportional relationships to the Pythagorean Theorem, students will gain a strong foundation in math through this unit. Mastering these knowledge points will not only equip studentswith essential mathematical skills but also enable them to approach real-world problems with confidence and analytical thinking.。
SAT数学的内容
SAT数学的内容美国教育考试服务中心（ETS）明确规定了SAT数学试题的出题范围。
表5-1-2是它的英文原文，表5-1-3是它的译文。
尽管表中提到了一些吓人的字眼，如数论，概率，统计，集合，想象题等，实际上是一些初级的的知识，国内高中学到的内容是足够的了，无需特别补学这些知识。
至于其他内容，对中国学生来说更是小菜一碟，不在话下。
由此可见就内容而言，中国学生所掌握的知识足以应对SAT数学考试，故在此不再赘言。
表5-1-2 SAT数学考试的出题范围（英文）Number and Operation Arithmetic word problems (including percent, ratio, and proportion)Properties of integers (even, odd, prime numbers, divisibility, etc.)Rational numbersSets (union, intersection, elements) Counting techniquesSequences and series (including exponential growth)Elementary number theoryAlgebra and Functions Properties of exponentsAlgebraic word problemsSolutions of linear equations and inequalities Systems of equations and inequalities Quadratic equationsRational and radical equationsEquations of linesAbsolute valueDirect and inverse variationConcepts of algebraic functionsGeometry and Measurement Area and perimeter of a polygonArea and circumference of a circleVolume of a box, cube, and cylinder Pythagorean Theorem and special properties of isosceles, equilateral, and right triangles Properties of parallel and perpendicular lines Coordinate geometryGeometric visualizationSlopeSimilarityTransformationsData Analysis, Statistics, and Probability Data interpretation (tables and graphs) Descriptive statistics (mean, median, and mode)Probability表5-1-3 SAT数学考试的出题范围（中文）数和运算算数应用题（包括百分数，比率和比例）整数性质（偶数，奇数，质数，整除性等等）有理数集合（合集，交集，元素）计数技术数列和级数（包括几何增长）数论初步代数和函数指数的性质代数应用题求解线性方程和线性不等式方程组和不等式组一元二次方程有理和无理方程直线方程绝对值变量的正反关系代数函数的概念几何和测量多边形的面积和周长园的面积和周长立方体，正方体和圆柱体的体积勾股定理和等腰，等边，直角三角形的性质平行线，垂直线的性质解析几何初步几何想象题斜率相似转换数据分析，统计和概率数据描述（表和图）统计知识（平均值，中位值，众数）概率三、SAT数学试题的特点SAT数学是中国考生最有可能得满分的一个部分。
SAT数学讲义：几何
SAT数学讲义：几何
备考SAT数学，同学们首先要对SAT数学有一个全面的了解，对于考试中出现的各个知识点，有充分的了解。
今天为大家整理了了比较全的SAT数学讲义，帮助大家全面的备考SAT数学考试。
几何(Geometry and Measurement)
包含线角、三角形(等边、等腰、直角)、四边形 (平行四边行、矩形、正方形)的面积和周长、正多边形(内角和、周长、面积)、圆、立体几何、坐标系、图形平移。
( l )欧几里德几何
补角supplementary angle 、余角 complementary angle 、同位角 corresponding angle 、内错角 alternative angle 、同旁内角 same-side interior angles ;
三种三角形、三种四边形、正多边形内角和。
acute triangle，obtuse triangle， rectangular triangle
( 2 )解三角形
特殊角的三角值、勾股定理Pythagorean theorem
( 3 )圆
直径半径、面积、周长、弧长直线与圆相切
radius，area，perimeter，arc length
( 4 )立体几何
圆、圆柱、圆锥、棱锥、棱柱等图形的半径、表而积、体积。
sphere，cylinder，cone，pyramind，rprism
( 5 )坐标系
平面直角坐标系、两点间冲、中点公式。
( 6 )图形平移
左加右减。
SAT数学知识点总结
SAT数学知识点总结青岛新航道学校为大家整理的SAT数学知识点，非常详细，按照相应的知识体系对知识点进行了一些梳理。
大家在备考SAT数学考试的时候可以根据自己的实际情况，对相应的内容进行整理和借鉴。
VIII TRIGONOMETRYA. Trigonometry of the right triangle1 Definitions of the six functions2 Relations of the functions of the complementary angles3 Reciprocal relations among the functions4 Variations in the functions of acute angles5 Pythagorean and quotient relations6 Functions of 30°, 45°, and 60°7 Applications of the functions to right triangle problemsB. Trigonometric functions of the general angle1 Generating an angle of any size2 Radians and degrees3 Using radians to determine arc length4 Definitions of the functions of an angle5 Signs of the functions in the four quadrants6 Functions of the quadrantal angle7 Finding the value of functions of any angleC Identities and equations1 Difference between identities in equations2 Proving identities3 Solving linear trigonometric functions4 Solving trigonometric quadratic equationsD Generalized trigonometric relationships1 Functions of the sum of two angles2 Functions of the difference of two angles3 Functions of the double angle4 Functions of the half angleE Graphs of trigonometric functions1 Graphs of the sine, cosine, and tangent curves2 Properties of the sine, cosine, and tangent curves3 Definitions of amplitude, period, and frequency4 Solving trigonometric equations graphicallyF Solutions of oblique triangles1 Law of sines2 Law of cosines3 Using logarithms to solve oblique triangle problems4 Vector problems—parallelogram of forces5 Navigation problemsIX MISCELLANEOUS TOPICSA. Complex numbers1 Meaning2 Operationsa) Addition and subtractionb) Multiplication and divisioni Powers of iii Complex conjugate3 Complex roots of quadratic equationsB Number Bases1 Converting from base 10 to other bases2 Converting from other bases to base 103 Operations in other basesC Exponents and logarithms1 Meaning of logarithms2 Computation with exponents and logarithms3 Equations4 Graphs of exponential and logarithmic functionsD Binary operations1 Definition of binary operations2 Properties of binary operations3 Application to modular arithmeticE Identity and inverse elements1 Addition2 Multiplication3 Other operations。
一维奇异p-Laplacian三点边值问题正解的存在性
一维奇异p-Laplacian三点边值问题正解的存在性白杰;祖力【摘要】利用非线性Leray-Schauder抉择定理和锥不动点定理,在假设条件下证明一维非线性奇异p-Laplacian三点边值问题解的存在性.结果表明,在区间(O,1]上至少存在一个正解.%By means of nonlinear Leray-Schauder alternative theorem and fixed point theorem in cones, thernauthors proved the existence of the solutions for one-dimensional singular p-Laplacian three-point boundaryrnvalue problems under assumptive conditions. There is at least one positive value in the interval from zero tornone.【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2012(050)004【总页数】7页(P621-627)【关键词】Leray-Schauder抉择定理;锥不动点定理;奇异边值问题;正解的存在性【作者】白杰;祖力【作者单位】东北师范大学人文学院信息技术学院,长春130117;长春大学理学院,长春130022;东北师范大学数学与统计学院,长春130024【正文语种】中文【中图分类】O175.140 引言关于一维p-Laplacian边值问题的研究目前已有许多结果[1-10]. 翁世有等[8]利用Schauder不动点原理和非线性Leray-Schauder抉择定理建立了一维p-Laplacian奇异边值问题解的一些存在性原则; Agarwal等[11-12]利用Leray-Schauder抉择定理得到了p=2时正解的存在性.考虑如下奇异边值问题:(1)其中: Φ(s)=s; p>1; q(t)在t=0处有奇性; 非线性项f可能在u=0 处有奇性. 本文应用文献[11-12]的方法, 证明p>1时问题(1)存在正解.1 预备知识假设：(H1) q(t): (0,1)→(0,∞)连续, 并且存在0≤α<p-1, 使得tαq(t)dt<∞成立;(H2) f(u)=g(u)+h(u), 其中: g>0在(0,∞)上连续且单调不增; h≥0在[0,∞)上连续; 且h/g在(0,∞)上单调不减;(H3) 存在一个常数r>0, 使得(2)成立, 其中Φ-1(u) ∶=sgn u是Φ(u)的反函数.例如, 当α∈(a-1,p-1)∩[0,p-1)时, 函数q(t)=t-a(0<t<1, 0≤a<p)满足条件(H1). 注1 容易验证条件(H1)表明若函数u(t)满足下列条件, 则u(t)是问题(1)的一个正解:1) u∈C[0,1]∩C1(0,1];2) 对任意的t∈(0,1], 有u(t)>0, 并且u(0)=0, u(1)=u(ξ), 0<ξ<1;3) Φ(u′(t))在(0,1)上一致绝对连续, 且(Φ(u′))′+q(t)f(u(t))=0, 0<t<1.定义1[13] 设X为实Banach空间, K是X中的闭凸子集, 若K满足下列条件, 则称K是X中的闭锥(简称锥):1) 若x∈K, λ≥0, 则λx∈K;2) 若x∈K, -x∈K, 则x=0.引理1(非线性Leray-Schauder抉择定理)[14] 假设K为Banach空间E的一个凸集, Ω为K的一个相对开子集, 0∈Ω, 映射为一个紧算子, 则下列条件必有一个成立:1) A在上有一个不动点;2) 存在x∈∂Ω和0<λ<1, 使得x=λA(x).定义C[0,1]中锥K为: K ∶={u∈C[0,1]: u(t)是非负的凹函数}.引理2 令h(t): (0,1)→(0,∞)连续, 且存在0≤α<p-1, 使得tαh(t)dt<∞, 则(3)存在唯一的正解V∈C[0,1]∩C1(0,1].证明：先证解的存在性.当0<t≤1时, 设显然, 由注1知, y(t)在(0,1]上连续严格增, 且y(ξ)<0<y(1). 因此, y(t)在(0,1)上只有一个零点. 令σ是y(t)在(0,1)上的唯一零点. 则令(4)则V在(0,1]上有定义, 且在(0,1]上V(t)>0. 进一步, 有(5)由(H2)知, 对0<t≤σ, 有则V(0)=0.类似可得V(1)=V(ξ). 因此, V(t)在[0,1]上连续, 且V(0)=0, V(1)=V(ξ); [Φ(V′(t))]′=-h(t), t∈(0,1).由比较原理易证唯一性. 证毕.令n≥4是一个固定的自然数. 对每个u∈K, 考虑如下问题:(6)其中F(u)=g*(u)+h(u), 满足注2 g*(u)≤g(u), ∀u∈(0,∞).由引理2, 可得:引理3 对每个固定的u∈K, 边值问题(6)存在唯一的解:w(t)=(Ψu)(t), w∈K,其中(7)σu∈(0,1)为如下方程在0≤τ≤1时的唯一解:对u∈K, 由w和Ψ的定义知:1)2) 在(0,1)中, (Φ(w′(t)))′=-q(t)F(u(t)), 且w(0)=1/n, w(1)=w(ξ);3) w=Ψu∈K, ‖w‖=w(σu).表明w(t)是问题(6)的一个解, 且为定义在[0,1]上的凹函数.类似文献[7]中引理2.6～引理2.9的证明方法, 可得下列引理.引理4 令wi(t)是F=Fi(i=1,2)时问题(6)的一个解. 如果F1≤F2, 则w1(t)≤w2(t).引理5 设[a,1]⊂(0,1]是一紧区间, 且令w(t)是F(u)≤M时问题(6)的一个解, 则w′(t)≤C(a,M), a≤t≤1.其中: M是一个正常数; C(a,M)是一个与a,M有关的正常数.注3 设w(t)是F(u)≤M时问题(6)的一个解, 则w(t)≤1/n+VM(t), 即(Ψu)(t)≤1/n+VM(t).注4 设w(t)是F(u)≥m时问题(6)的一个解, 则w(t)≥1/n+Vm(t), 即(Ψu)(t)≥1/n+Vm(t).引理6 对任意有界闭子集Ω⊂K, 集合Ψ(Ω)在[0,1]上等度连续.引理7 对任意的有界闭子集Ω⊂K, 映射Ψ: Ω→K是连续的.综合引理3～引理7, 可得:引理8 Ψ: K→K是全连续的.2 主要结果定理1 假设条件(H1)～(H3)成立, 则在区间(0,1]上, 系统(1)至少存在一个解u∈C[0,1]∩C1(0,1], 满足u>0, 且‖u‖<r.证明: 先用引理1证明解的存在性. 选择ε>0, 且ε<r, 使得(9)选择n0∈{1,2,…}, 使得1/n0<ε. 令N+={n0,n0+1,…}.下面证明边值问题:(10)在(0,1]上有一个解: 且‖un‖<r.∀n∈N+, 为证式(10)有一个解, 需考虑如下边值问题:(11)其中F的定义见式(6).固定n∈N+. 定义为式(7), 式(7)中σu∈(0,1)为如下方程的唯一解:由引理8, 可得是全连续的.下面证明u≠λΨu, λ∈(0,1), u∈∂Ωr.(12)假设式(12)不成立, 即存在一个λ∈(0,1)和u∈∂Ωr, 使得u=λΨu, 则有(13)显然存在σn∈(0,1), 使得在(0,σn)上, u′(t)≥0; 在(σn,1)上, u′(t)≤0, 且u(σn)=‖u‖=r. 再注意到F(u(t))≤g(u(t))+h(u(t)), t∈(0,1),则当z∈(0,1)时,(14)对式(14)从t(0<t≤σn)到σn积分, 得(15)则有(16)再从0到σn积分得(17)即(18)因此(19)这与条件(9)矛盾, 于是式(12)成立.由引理1可知Ψ有一个不动点即1/n≤‖un‖≤r(注意到, 如果‖un‖=r, 则与式(14)～(19)的证明同理可得矛盾). 因为un≥1/n, 所以un(t)也是问题(10)的一个解. 由(H2), 当r>0时,g(un(t))≥g(r), f(un)=h(un)+g(un)≥g(r).则由注4, 可得(20)注5 注意到在区间(0,1]上, Vg(r)(t)>0, 则un(t)>0, t∈(0,1].下面证明{un}n∈N+在[0,1]上一致有界且等度连续. 由式(14)(用un代替u), 可得(21)因为在[0,1]上, un(t)≥1/n, 则在(0,σn)上存在σn∈(0,1), 使得而在(σn,1)上, 且un(σn)=‖un‖≤r.对式(21)从t(0<t<σn)到σn积分得(22)下面证明存在a0>0, 使得a0<inf{σn: n∈N+}≤1.(23)如果式(23)不成立, 则存在N+的子列S, 使得当S中的n→∞时, σn→ 0. 对式(22)从0到σn积分得(24)其中n∈S. 因为当n→∞时, σn→ 0, 则由式(24)可得, 当n→∞时, un(σn)→ 0. 又因为un在[0,1]上σn处取得最大值, 所以当n→∞时, C[0,1]中的函数un→ 0. 这与式(20)矛盾. 表明(25)其中W(t)=q(z)dz. 由注2知, Φ-1(W)∈L1[0,1].对式(21)从σn(σn<t<1)到t积分得当σn≤t≤1时, 有(26)则式(25),(26)表明, 当t∈(0,1)时,(27)定义I: [0,∞)→[0,∞)为I(z) 注意到I: [0,∞)→[0,∞)是单调增的映射, 且I(∞)=∞, 这是因为g(u)>0在(0,∞)上单调不减, 且对任意的B>0, I在[0,B]上连续.{I(un)}n∈N+在[0,1]上一致有界且等度连续, 其等度连续性可从下式得到(这里t,s∈[0,1]):由不等式(28)、 I-1的一致连续性及un(t)-un(s)=I-1(I(un(t)))-I(un(s))可知{un}n∈N+在[0,1]上一致有界且等度连续.由Arzela-Ascoli定理, N+存在一个子列N⊂N+, 使得当n∈N, n→∞时, 存在u∈C[0,1], 使得un在[0,1]上一致收敛于u. 则由式(20)知, 在[0,1]上,un(t)≥Vg(r)(t). 特别地, 在(0,1]上, u(t)>0.固定t∈(0,1], 有(29)由式(26), 有则有一个收敛子列; 为方便, 仍用表示该子列, 并且令r0∈R表示其极限. 则对上面固定的t∈(0,1], 在N上, 令n→∞(注意到q f在紧子区间[t,1]×(0,r]上一致连续)得(30)t取遍(0,1]可得因此r0=u′(1), 从而有(Φ(u′))′+q(t)f(u(t))=0, 0<t<1, u(0)=u(1)-u(ξ)=0.最后易证‖u‖<r(注意到如果‖u‖=r, 与式(14)～(19)的证明同理可推出矛盾). 从而证明了问题(1)至少有一个正解u(t)∈C[0,1]∩C1(0,1], 且‖u‖<r. 证毕.3 应用实例考虑奇异边值问题:(31)其中: 0≤m<p; σ>0; α>0; β>p-1.设则b0=σ1/(p-1)b1.应用定理1可知, 如果存在r>0满足(32)则问题(31)存在一个正解.设则选择r=x0, 则式(32)成立. 显然, 定理1中的(H1)～(H3)成立. 因此, 问题(31)存在一个解u∈C[0,1]∩C1(0,1], 使得在(0,1]上, u>0且‖u‖<r=x0.参考文献【相关文献】[1] XU Xian. Multiplicity Results for Positive Solutions of Some Semi-position Three-Point Boundary Value Problems [J]. J Math Anal Appl, 2004, 291(2): 673-689.[2] SUN Jing-xian, XU Xian, O’Regan D. Nodal Solutions for m-Point Boundary Value Problems Using Bifurcation Methods [J]. Nonlinear Anal: Theory, Method & Applications, 2008, 68(10): 3034-3046.[3] Gupta C P. Existence and Uniqueness Theorems for the Bending of an Elastic Beam Equations [J]. Appl Anal, 1988, 26(4): 289-304.[4] Gupta C P. Solvability of a Three-Point Nonlinear Boundary Value Problem for a Second Order Ordinary Differential Equation [J]. J Math Anal Appl, 1992, 168(2): 540-551.[5] KONG Ling-bin, WANG Jun-yu. Multiple Positive Solutions for the One-Dimensional p-Laplacian [J]. Nonlinear Anal: Theory, Method & Applications, 2000, 42(8): 1327-1333. [6] Agarwal R P, O’Regan D. Twin Solutions to Sin gular Dirichlet Problems [J]. J Math Anal Appl, 1999, 240(2): 433-445.[7] JIANG Da-qing, XU Xiao-jie. Multiple Positive Solutions to a Class of Singular Boundary Value Problems for the One-Dimensional p-Laplacian [J]. Comput Math Appl, 2004,47(4/5): 667-681.[8] WENG Shi-you, GAO Hai-yin, ZHANG Xiao-ying, et al. Existence Principles for Singular Boundary Value Prolems of One Dimension p-Laplacian [J]. Journal of Jilin University: Science Edition, 2006, 44(3): 351-356. (翁世有, 高海音, 张晓颖, 等. 一维p-Laplacian奇异边值问题的存在性原则 [J]. 吉林大学学报: 理学版, 2006, 44(3): 351-356.)[9] YUAN Cheng-jun, WEN Xiang-dan, MENG Qing-yuan. Existence and Uniqueness of Positive Solutions of Fourth-Order Nonlinear Singular Discrete Boundary Value Problems with p-Lapacian Operator [J]. Journal of Northeast Normal University: Natural Science Edition, 2010, 42(1): 5-9. (苑成军, 文香丹, 孟庆元. 奇异四阶p-Lapacian差分方程边值正解的存在唯一性 [J]. 东北师大学报: 自然科学版, 2010, 42(1): 5-9.)[10] YUAN Cheng-jun, WEN Xiang-dan. Existence and Uniqueness of Positive Solutions for Fourth-Order Nonlinear Singular Continuous Boundary Value Problems with p-Lapacian Operator [J]. Journal of Natural Science of Heilongjiang University, 2009, 26(2): 190-193. (苑成军, 文香丹. 奇异四阶p-Lapacian微分方程边值正解的存在惟一性 [J]. 黑龙江大学自然科学学报， 2009, 26(2): 190-193.)[11] Agarwal R P, O’Regan D. Existence Theory for Single and Multiple Solutions to Singular Positone Boundary Value Problems [J]. J Differential Equations, 2001, 175(2): 393-414.[12] Agarwal R P, O’Regan D. Twin Solutions to Singular Boundary Value Problems [J]. Proc Amer Math Soc, 2000, 128: 2085-2094.[13] 钟承奎, 范先令, 陈文源. 非线性泛函分析引论 [M]. 兰州: 兰州大学出版社, 1998.[14] Agarwal R P, O’Regan D. Nonlinear Superlinear Singular and Nonsingular Second Order Boundary Value Problems [J]. J Differential Equations, 1998, 143(1): 60-95.。
SAT考试2024数学历年题目全解
SAT考试2024数学历年题目全解SAT考试是一项全球性的标准化考试，旨在评估学生在阅读、写作和数学方面的能力。
数学部分是SAT考试的一个重要组成部分，涵盖了各种数学概念和技巧。
本文将为您提供2024年SAT数学部分的历年题目全解，帮助您更好地应对这一考试。
第一题：题目：求解以下方程：3x + 5 = 20解析：要求解方程3x + 5 = 20，我们首先将5从等式两边减去，得到3x = 15。
然后，我们将方程两边都除以3，即x = 5。
因此，方程的解为x = 5。
第二题：题目：计算以下比例的值：5:8 = x:40解析：要计算比例5:8与x:40的值，我们可以采取交叉乘法的方法。
将5乘以40，并将结果除以8，即可求得x的值。
计算过程如下：5 * 40 / 8 = 200 / 8 = 25因此，比例5:8与x:40的值为25。
第三题：题目：已知一个等边三角形的边长为12，计算其面积。
解析：一个等边三角形的边长为12，则其高可以通过勾股定理求得。
根据勾股定理，我们有：高的平方= 边长的平方- 底边的一半的平方。
设高为h，则有 h^2 = 12^2 - (12/2)^2= 144 - 36= 108因此，高h = √108 = 6√3由于等边三角形的高等于边长的一半乘以根号3，所以面积S可以计算为：S = 1/2 * 12 * 6√3= 6 * 6√3= 36√3因此，该等边三角形的面积为36√3。
第四题：题目：在一个长方形花坛中，长度是宽度的3倍，已知宽度为2米，计算花坛的面积。
解析：我们知道长方形花坛的面积可以通过长度乘以宽度来计算。
已知宽度为2米，则长度为3 * 2 = 6米。
因此，花坛的面积为2 * 6 =12平方米。
通过以上题目的解析，我们可以看到SAT数学部分考察了各种数学概念和技巧，包括方程的求解、比例的计算、勾股定理的应用以及长方形面积的计算等。
熟练掌握这些数学知识，并能够灵活运用于实际问题的解决中，将有助于您在SAT考试中取得更好的成绩。
SAT数学知识点
SAT 数学知识点一Number and Operations Review 一、Properties of integers知道下列说法表示的内容：1. Integers consist of the whole numbers and their negatives (including zero).2. Integers extend infinitely in both negative and positive directions.3. Integers do not include fractions or decimals.4. Negative integers5. Positive integers6. The integer zero is neither positive nor negative.7. odd numbers（奇数）and even numbers（偶数）8. Consecutive integers9. Addition of integers(奇数偶数的加法规则)10. Multiplication of integers（奇数偶数的乘法规则）二、Arithmetic word problems（算术题）三、Number lines（数轴）四、Square and square roots（平方和平方根）五、Fractions and rational numbers（分数与有理数）六、Elementary number theory☆Factors, multiples, and remainders☆Prime numbers七、Ratios, proportions, and percents八、Sequences九、Sets(union, intersection, elements)十、Counting problems Counting problems involve figuring out how many ways you can select or arrange members of groups, such as letters of the alphabet, numbers or menu selections.☆Fundamental counting problems分步完成事件和分类完成事件发生的可能性☆Permutations and combinations （排列组合）基本排列组合理论十一、Logical reasoningThe SAT doesn’t include1.Tedious or long computations2.Matrix operations1.2.3.4.5. 6.7.8.9.10.11.12.13. 14.SAT数学知识点二Algebra and Functions Review Many math questions require knowledge of algebra. This chapter gives you some further practice. You have to manipulate and solve a simple equation for an unknown, simplify and evaluate algebraic expressions, and use algebraic expressions, and use algebraic concepts in problem-solving situations.For the math questions covering algebra and functions content, you should be familiar with all of the following basic skills and topics:一、Operations on algebraic expressions二、Factoring三、Exponents四、Evaluating expressions with exponents and roots五、Solving equations☆Working with “unsolvable” equations☆Solving for one variable in terms of another☆Solving equations involving radical expressions六、Absolute value 七、Direct translation into mathematical expressions八、Inequalities九、Systems of linear equations and inequalities十、Solving quadratic equations by factoring 十一、Rational equations and inequalities 十二、Direct and inverse variation十三、Word problems十四、Functions☆Function notation and evaluation☆Domain and range☆Using new definitions☆Functions as models☆Linear functions: their equations and graphs☆Quadratic functions: their equations and graphs☆Qualitative behavior of graphs and functions☆Translations and their effects on graphsand functionsThe SAT doesn’t include:一、Solving quadratic equations thatrequire the use of the quadraticformula二、Complex numbers三、Logarithms1.2.3.4. 5.6.7.8.9.10.SAT 数学知识点三Geometry and Measurement Review Concept you should to knowFor the mathematics questions covering geometry and measurement concepts, you should be familiar with all of the following basic skills, topics, and formulas:一、Geometric notation二、Points and lines三、Angles in the plane四、Triangles(including special triangles)☆Equilateral triangles☆Isosceles triangles☆Right triangles and the Pythagorean theorem ☆30º-60º-90ºtriangles☆45º-45º-90ºtriangles☆3-4-5 triangles☆Congruent triangles☆Similar triangles☆The triangle inequality五、Quadrilaterals☆Parallelograms☆Rectangles☆Squares六、Areas and Perimeters☆Areas of squares and rectangles☆Perimeters of squares and rectangles☆Area of triangles☆Area of Parallelograms七、Other polygons☆Angles in a polygon☆Perimeter☆Area八、Circles☆Diameter☆Radius☆Arc☆Tangent to a circle☆Circumference☆Area九、Solid geometry☆Solid figures and volumes☆Surface area十、Geometric perception十一、Coordinate geometry☆Slopes, parallel lines, and perpendicular lines☆The midpoint formula☆The distance formula十二、TransformationsThe SAT doesn’t include:一、Formal geometric proofs二、Trigonometry三、Radian measure1.2.3.4.5.6. 7.8.9.SAT 数学知识点四Data Analysis, Statistics andProbability ReviewFor the math questions covering data analysis, statistics and probability concepts, you should be familiar with all of the following basic skills and topics:一、Data interpretation二、Statistics☆Arithmetic mean☆Median☆Mode☆Weighted average☆Average of algebraic expression☆Using average to find missing numbers三、Elementary probability四、Geometric probabilityThe SAT doesn’t include:四、Computation of standard deviation 1.2.3.4.5. 6.7.8.Word Problems1.2.3.4. 5-75.6.7.1112。
sat数学考纲
sat数学考纲SAT（Scholastic Assessment Test）是学术能力评估测试，是一种标准化考试，是美国大学入学考试。
国内的学生要申请美国的本科，都是需要参加SAT考试的，那么SAT考试内容有哪些呢？下面为大家详细的介绍介绍。
SAT考试有两种：普通考试（General Test）和学科考试（Subject Test）。
SAT普通考试包括数学和证据阅读和写作。
学科考试包括数学、科学、语言、历史和英语。
今天我们主要介绍普通考试的考试内容。
1、SAT数学教学大纲代数学•线性函数的图形表示•有理系数线性方程•二元线性不等式及其系统•线性方程组(无解、无限解或有限解)数据分析和问题解决•比率和比例•单位换算•使用散点图的直线或曲线方程•计算条件频率和条件概率的双向表•变量的关联或事件的独立性•总体参数的估计•统计学中平均值、中位数、众数、极差和标准差的计算•评估报告以检查数据收集方法的适当性高等数学•有理系数二次方程•确定表达式的形式•多项式方程(减法、乘法、加法和除法)•多项式的零点和因子•两个变量之间的非线性关系•函数符号•通过重新排列公式或方程式来隔离变量。
2、SAT写作和语言教学大纲SAT写作和语言通用考试大纲也可能包含评估考生基本技能的图表。
这些主题分为以下几类:•科学•职业•历史和艺术•社会科学SAT写作和语言教学大纲根据句子结构、动词时态、标点符号、主谓一致、语法用法、加强/削弱论点等提出问题。
在SAT写作大纲中，有一些包含段落的题型。
SAT考试大纲的写作部分包括: •句子结构、动词时态、标点符号、平行结构、语法用法、主谓一致等。
•加强/削弱论点•风格、文本或语气的用词选择•改进可读性的结构变化3、SAT阅读教学大纲SAT阅读教学大纲由5段不同来源的文章组成。
两段是科学的，一段基于美国或世界文学的经典或当代文本。
SAT课程印度阅读的其他段落包括经济学、心理学、社会学等。
以下是SAT阅读教学大纲中的题型：•一个短语在文章中的作用•作者的观点、态度、风格和语气•分析信息图表数据，如图形、表格、图表等•学习文章的中心主题或主要思想•支持作者陈述的证据或论点4、SAT作文(可选)SAT作文部分是可选的。
最新SAT数学知识讲解：三角形
最新SAT数学知识讲解：三角形这篇关于最新SAT数学知识讲解：三角形，是笔者特地为大家整理的，希望对大家有所帮助！30-60-90 TrianglesThe guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles have angles of ,, and. What’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x . The sides of every 30-60-90 triangle will follow this ratio of 1:: 2 . This constant ratio means that if you know the length of just one side in the triangle, you’ll immediately be able to calculate the lengths of all the sides. If, for example, you know that the side oppositethe 30º angle is 2 meters long, then by using the 1:: 2 ratio, you can work out that the hypotenuse is 4 meters long, and the leg opposite the 60º angle is 2meters.And there’s another amazing thing about 30-60-90 triangles. Two of these triangles joined at the side opposite the 60º angle will form an equilateral triangle. Here’s why you need to pay attention to this extra-special feature of 30-60-90 triangles. If you know the side length of an equilateral triangle, you can figure out the triangle’s height: Divide the side length by two and multiply it by. Similarly, if you drop a “perpendicular bisector” (this is the term the SAT uses) from any vertex of an equilateral triangle to the base on the far side, you’ll have cut that triangle into two30-60-90 triangles.Knowing how equilateral and 30-60-90 triangles relate is incredibly helpful on triangle, polygon, and even solids questions on the SAT. Quite often, you’ll be able to break down these large shapes into a number of special triangles, and then you can use the side ratios to figure out whatever you need to know.45-45-90 TrianglesA 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 30-60-90 triangle, the lengths of the sides of a 45-45-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x : Know this 1: 1:ratio for 45-45-90 triangles. It will save you time and may even save your butt.Also, just as two 30-60-90 triangles form an equilateral triangles, two 45-45-90 triangles form a square. We explain the colossal importance of this fact when we cover polygons a little later in this chapter.Similar TrianglesSimilar triangles have the same shape but not necessarily the same size. Or, if you prefer more math-geek jargon, two triangles are “similar” if theratio of the lengths of their corresponding sides is constant (which you now know means that their corresponding angles must be congruent). Take a look at a few similar triangles: As you may have assumed from the figure above, the symbol for “is similar to” is ~. So, if triangle ABC is similar to triangle DEF, we write ABC ~ DEF.There are two crucial facts about similar triangles.Corresponding angles of similar triangles are identical.Corresponding sides of similar triangles are proportional.For ABC ~ DEF, the corresponding angles areThe corresponding sides are AB/DE = BC/EF =CA/FD.The SAT usually tests similarity by presenting you with a single triangle that contains a line segment parallel to one base. This line segment creates a second, smaller, similar triangle. In the figure below, for example, line segment DE is parallel to CB, and triangle ABC is similar to triangle AE. After presenting youwith a diagram like the one above, the SAT will ask a question like this:If= 6 and=, what is?。
SAT考试2024数学历年题目精讲
SAT考试2024数学历年题目精讲在本篇文章中，我们将重点讲解SAT考试2024年数学部分的历年题目。
我们将按照题目类型进行分类，并为每个题型提供详细的解答和解题技巧，帮助考生更好地应对这些题目。
一、单选题1. 题目描述：某汽车展厅共展出了150辆汽车，其中的三分之一是SUV车型，四分之一是轿车车型，其余的是其他车型。
问展厅中轿车车型的数量是多少？解答与技巧：首先，计算出SUV车型的数量：150 * (1/3) = 50辆。
然后，计算出其他车型的数量：150 - 50 - 150 * (1/4) = 50辆。
所以，轿车车型的数量是50辆。
2. 题目描述：某商场举办了一次打折活动，原价100元的商品现在只需80元购买。
如果小明购买了3件该商品，他需要支付多少钱？解答与技巧：首先，计算出每件商品的折扣金额：100 - 80 = 20元。
然后，计算出小明需要支付的金额：3 * 20 = 60元。
所以，小明需要支付60元。
二、多选题1. 题目描述：以下哪些数是正整数？（A）-1（B）0（C）1（D）2解答与技巧：在SAT考试中，如果题目要求选择多个选项，我们需要仔细审题。
在这个题目中，需要选择正整数，所以选项B和A都不是正整数。
所以正确答案是（C）和（D）。
2. 题目描述：以下哪些图形具有对称性？（A）正方形（B）长方形（C）圆形（D）三角形解答与技巧：我们需要判断每个选项是否具有对称性。
在这个题目中，正方形和圆形都具有对称轴，所以正确答案是（A）和（C）。
三、填空题1. 题目描述：若a + a^-1 = 5，求a^2 + a^-2的值。
解答与技巧：首先，我们可以对等式两边进行平方操作，得到a^2+ 2 + a^(-2) = 25。
然后，我们需要解方程，将等式左边与右边的常数项进行抵消，得到a^2 + a^(-2) = 23。
2. 题目描述：某比赛共有10个选手参加，其中3个选手退出比赛，剩余的选手中将决出第一名、第二名和第三名。
常用的SAT数学词汇中英文对照
常用的SAT数学词汇中英文对照第一篇：常用的SAT数学词汇中英文对照常用的SAT数学词汇中英文对照常用的SAT数学词汇中英文对照：代数部分1、有关基本运算：add，plus加subtract减difference差multiply,times乘product积divide除divisible可被整除的dividedevenly被整除dividend被除数divisor因子，除数quotient商remainder余数factorial阶乘power乘方radicalsign,rootsign根号roundto四舍五入tothenearest四舍五入2.有关集合union并集proper subset真子集solution set解集3.有关代数式、方程和不等式algebraic term代数项like terms,similar terms同类项numerical coefficient数字系数literal coefficient字母系数inequality不等式4.有关分数和小数5.基本数学概念triangle inequality三角不等式 range值域 original equation原方程 equivalent equation同解方程等价方程 linear equation线性方程(e.g.5x+6=22)proper fraction真分数improper fraction假分数 mixed number带分数 vulgar fraction，common fraction普通分数 simple fraction简分数 complex fraction繁分数 numerator分子denominator分母(least)common denominator(最小)公分母quarter四分之一 decimal fraction纯小数 infinite decimal无穷小数recurring decimal循环小数 tenthsunit十分位arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数exponent指数，幂base乘幂的底数,底边cube立方数，立方体square root平方根cube root立方根common logarithm常用对数digit数字constant常数variable变量inversefunction反函数complementary function余函数linear一次的，线性的factorization因式分解第二篇：数学常用词汇中英文对照数学常用词汇中英文对照代数部分1、有关基本运算：add，plus加subtract减difference差multiply,times乘product积divide除divisible可被整除的dividedevenly被整除dividend被除数divisor因子，除数quotient商remainder余数 factorial阶乘power 乘方radicalsign,rootsign根号roundto四舍五入tothenearest四舍五入2.有关集合union并集proper subset真子集solution set解集3.有关代数式、方程和不等式algebraic term代数项like terms,similar terms同类项 numerical coefficient数字系数literal coefficient字母系数 inequality不等式triangle inequality三角不等式 range值域original equation原方程equivalent equation同解方程等价方程linear equation线性方程4.有关分数和小数proper fraction真分数 improper fraction假分数mixed number带分数 vulgar fraction，common fraction普通分数 simple fraction简分数 complex fraction繁分数 numerator分子 denominator分母(least)common denominator(最小)公分母 quarter四分之一decimal fraction纯小数infinite decimal无穷小数recurring decimal循环小数 tenthsunit十分位5.基本数学概念arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数exponent指数，幂base乘幂的底数,底边 cube立方数，立方体 square root平方根 cube root立方根common logarithm常用对数 digit数字 constant常数 variable 变量inversefunction反函数complementary function余函数linear一次的，线性的factorization因式分解 absolute value绝对值，round off四舍五入6.有关数论natural number自然数positive number正数negative number负数odd integer,odd number奇数 even integer,even number偶数integer,whole number整数positive whole number正整数negative whole number负整数 consecutive number连续整数rea lnumber,rational number实数,有理数irrational(number)无理数 inverse倒数composite number合数 prime number质数 reciprocal倒数common divisor公约数 multiple倍数(least)common multiple(最小)公倍数(prime)factor(质)因子common factor公因子ordinaryscale,decimalscale十进制 nonnegative非负的 tens十位 units个位 mode众数 median中数common ratio公比7.数列arithmetic progression(sequence)等差数列geometric progression(sequence)等比数列8.其它approximate近似(anti)clockwise(逆)顺时针方向 cardinal基数 ordinal序数directproportion正比distinct不同的estimation估计，近似parentheses括号 proportion比例 permutation排列 combination 组合 table表格trigonometric function三角函数 unit单位,位几何部分 1.所有的角alternate angle内错角corresponding angle同位角 vertical angle对顶角central angle圆心角 interior angle内角 exterior angle外角supplement aryangles补角complement aryangle余角adjacent angle邻角 acute angle锐角 obtuse angle钝角 right angle 直角 round angle周角 straight angle平角 included angle夹角2.所有的三角形equilateral triangle等边三角形scalene triangle不等边三角形isosceles triangle等腰三角形 right triangle直角三角形 oblique斜三角形inscribed triangle内接三角形3.有关收敛的平面图形，除三角形外semicircle半圆concentric circles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形 octagon八边形 nonagon九边形 decagon十边形 polygon多边形parallelogram平行四边形 equilateral等边形 plane平面square正方形，平方 rectangle长方形regular polygon正多边形 rhombus菱形 trapezoid梯形4.其它平面图形arc弧line,straight line直线line segment线段 parallel lines平行线 segment of a circle弧形5.有关立体图形cube立方体，立方数 rectangular solid长方体regular solid/regular polyhedron正多面体 circular cylinder圆柱体 cone圆锥 sphere球体 solid立体的6.有关图形上的附属物altitude高 depth深度 side边长circumference,perimeter周长 radian弧度surface area 表面积 volume体积arm直角三角形的股 cros ssection横截面 center of acircle圆心chord弦 radius半径angle bisector角平分线 diagonal对角线 diameter直径 edge 棱face of a solid立体的面 hypotenuse斜边included side夹边leg三角形的直角边medianofatriangle三角形的中线 base底边，底数opposite直角三角形中的对边 midpoint中点 endpoint端点vertex(复数形式vertices)顶点tangent切线的transversal截线intercept截距7.有关坐标coordinate system坐标系rectangular coordinate直角坐标系 origin原点 abscissa横坐标ordinate纵坐标 numberline数轴 quadrant象限 slope斜率complex plane复平面8.其它plane geometry平面几何 trigonometry三角学 bisect平分circumscribe外切 inscribe内切 intersect相交perpendicular垂直pythagorean theorem勾股定理 congruent全等的 multilateral 多边的1、单位类 cent美分penny一美分硬币 nickel5美分硬币 dime一角硬币 dozen 打（12个） score廿(20个) Centigrade摄氏 Fahrenheit华氏 quart夸脱gallon加仑(1gallon=4quart)yard码meter米micron 微米 inch英寸 foot英尺minute分(角度的度量单位，60分=1度)squaremeasure平方单位制 cubicmeter立方米pint品脱(干量或液量的单位)2.有关文字叙述题，主要是有关商业 intercalary year(leapyear)闰年(366天)common year平年(365天) depreciation折旧down payment直接付款discount打折margin利润profit利润 interest利息simple interest单利compounded interest复利 dividend红利 decrease to减少到 decrease by减少了 increase to增加到 increase by增加了 denote表示 list price标价 markup涨价 per capita每人 ratio比率retail price零售价 tie打平集合与简易逻辑集合（集）set非负整数集the set of all non-negative integers自然数集the set of all natural numbers 正整数集the set of all positive integers 整数集the set of all integers有理数集the set of all rational numbers 实数集the set of allreal numbers 元素element 属于belong to不属于not belong to 有限集finite set 无限集infinite set 空集empty set包含inclusion, include 包含于lie in 子集subset 真子集proper subset补集（余集）complementary set 全集universe 交集intersection 并集union偶数集the set of all even numbers 奇数集the set of all odd numbers 含绝对值的不等式inequality with absolute value 一元二次不等式one-variable quadratic inequality 逻辑logic逻辑联结词logic connective 或or 且and 非not 真true 假false 真值表truth table原命题original proposition 逆命题converse proposition 否命题negative proposition逆否命题converse-negative proposition 充分条件sufficient condition 必要条件necessary condition充要条件sufficient and necessary condition ……的充要条件是……… if and only if …函数函数function 自变量argument 定义域domain 值域range 区间interval闭区间closed interval 开区间open interval函数的图象graph of function 映射mapping 象image原象inverse image 单调monotone增函数increasing function减函数decreasing function 单调区间monotone interval 反函数inverse function 指数exponent n次方根n th root 根式radical 根指数radical exponent 被开方数radicand 指数函数exponential function 对数logarithm 常用对数common logarithm 自然对数natural logarithm 对数函数logarithmic function数列数列sequence of number 项term 通项公式the formula of general term 有穷数列finite sequence of number 无穷数列infinite sequence of number 递推公式recurrence formula 等差数列arithmetic progression，arithmetic series 公差common difference 等差中项arithmetic mean 等比数列geometric progression，geometric series 公比common ratio 等比中项geometric mean三角函数三角函数trigonometric function 始边initial side 终边terminal side 正角positive angle 负角negative angle 零角zero angle 象限角quadrant angle 弧度radian 弧度制radian measure 角度制degree measure 正弦sine 余弦cosine 正切tangent 余切cotangent 正割secant 余割cosecant 诱导公式induction formula 正弦曲线sine curve 余弦曲线cosine curve 最大值maximum 最小值minimum 周期period 最小正周期minimal positive period 周期函数periodic function 振幅amplitude of vibration 频率frequency 相位phase 初相initial phase 反正弦arc sine 反余弦arc cosine 反正切arc tangent 平面向量有向线段directed line segment数量scalar quantity向量vector 零向量zero vector 相等向量equal vector 共线向量collinear vectors 平行向量parallel vectors 向量的数乘multiplication of vector by scalar 单位向量unit vector 基底base 基向量basevectors 平移translation 数量积innerproduct 正弦定理sinetheorem 余弦定理cosinetheorem不等式算术平均数arithmetic mean 几何平均数geometric mean 比较法method of compare 综合法method of synthesis 分析法method of analysis直线倾斜角angle of inclination 斜率gradient点斜式point slope form 截距intercept斜截式 gradient intercept form 两点式two-point form 一般式general form 夹角included angle线性规划linear programming 约束条件constraint condition 目标函数 objective function 可行域feasible region 最优解optimal solution圆锥曲线曲线curve坐标法method of coordinate 解析几何analytic geometry 笛卡儿Descartes标准方程standard equation 一般方程general equation 参数方程parameter equation 参数parameter圆锥曲线point conic 椭圆ellipse焦点focus, focal points 焦距focal length 长轴major axis 短轴minor axis 离心率eccentricity 双曲线hyperbola 实轴real axis 虚轴imaginary axis 渐近线asymptote 抛物线parabola 准线directrix第三篇：保险业常用词汇中英文对照保险业常用词汇中英文对照第一部分：保险承保业务1、一般保险原理保险 insurance保险业insurance industry 保险业务结构business structures 保险业主体bodies in Insurance Industry 被保险人 insured 理赔claim settlement承保利润微薄underwriting profit is slight 承保能力insurance capacity承保业务underwriting operation 未决赔款储备金 reserve IBRN 大数法则 Law of Large Numbers营业额turnover市场缺陷 market imperfections社会保险social insurance个人代理人personal agent银行保险bank insurance直销straight pin佣金brokerage手续费commissioncharges 再保险reinsurance2、财产保险财产保险property insurance长尾long－tail长期护理保险long—term care insurance短尾short－tail综合赔付率combined ratio自留retention车险auto insurance财产保险property insurance责任保险liability insurance农业保险agriculturalinsurance巨灾保险catastrophe insurance意外伤害保险casualty accidentinsurance3、寿险词汇变额寿险 variable life insurance长寿保险insurance on last survivor定期寿险term insurance合同储蓄机构 contractual saving institutions红利 dividend疾病保险sickness insurance简易险 industrial insurance健康保险health insurance巨额医药费保险major medical" insurance policy可变保费寿险 flexible—premium life insurance利差损interest spread risk利率敏感型产品 interest—sensitive whole life赔期/期限indemnity period/limit普通寿险 ordinary life insurance人寿保险life insurance预定利率市场化Market-based Assumed Interest Rate团险group insurance限期缴费终身寿险 limited—payment whole life insurance 养老保险endowment insurance医疗保险insurance for medical care医疗费用保险medical expense insurance终身寿险whole life insurance准备金reserve伤残收入保险 disability income insurance失能 incapacity可变万能寿险Variable-universal life insurance可变年金Variable annuity评估价值Appraisal Value内含价值Embeded Value新业务价值Value of New Business税收优惠taxbreak税收递延型养老保险Deferred tax endowment insurance大病医疗保险Large quantity medical insurance分红保险Participating Insurance传统保险traditional insurance理财型保险financial planning oriented insurance投资连结寿险 unit—linked life insurance4、保险偿付能力风险资本要求risk—based capital固定最低成本要求 fixed minimum capital requirement保险偿付能力监管insurance solvency regulation保险监管信息系统 Insurance Regulatory Information System 财务分析跟踪系统 Financial Analysis Tracking System偿付能力Solvency偿付能力不足 insolvent认可资产admissible assets第二部分保险资产管理1、行业主体和监管保险资产管理insurance asset management保险法 insurance law保险资产管理公司 insurance asset management company中国人保集团People's Insurance Company of China(简称 PICC) 中国人保资产管理公司PICC Asset Management Company Limited慕尼黑再保险资产管理公司Munich ERGO Asset Management GmbH（简称MEAG）中国保监会China Insurance Regulatory Commission（简称CIRC）保险投资“新政”innovati on policies in insurance investment 可投资资产investable asset投资公司法案 Investment Company Act2、委托受托关系委托人 client受托人 trustee委托管理 mandate绝对收益absolute return相对收益 relative return3、组合管理大类资产配置 assets allocation战略资产配置Strategic Assets Allocation（简称SAA）战术资产配置 Tactic Assets Allocation（简称TAA）流动性管理 liquidity management资产负债管理 Asset-Liability Management（简称ALM）久期匹配duration matching现金流匹配cash flow matching一般账户 general account独立账户separate account4、类别投资固定收益投资fixed-income investment持有到期类资产held-to-maturity asset交易类资产 trading asset可供出售类资产 asset available for sale股票投资 equity investment基金投资fund investment另类投资 alternative investment境外投资 overseas investment基础设施债权计划infrastructure debt investment plan未上市企业股权投资private equity investment 不动产投资real estate investment利率互换Interest Rate Swap（简称IRS）均值一方差分析the mean-variance method5、第三方业务财富管理wealth management第三方投资管理the third-party asset management企业年金occupational annuity养老金 pension集合投资产品 collective investment fund第四篇：英文简历常用词汇中英文对照•英文简历常用词汇中英文对照应聘职位名称（中英文对照）Executive and Managerial(管理部分)Retail Store Manager 零售店经理Food Service Manager Executive Marketing Director市场行政总监HMO AdministratorAssistant Store Manager 商店经理助理Operations Manager Assistant Vice-President副总裁助理Production Manager Chief Executive Officer(CEO)首席执行官Property ManagerChief Operations Officer(COO)首席运营官Branch ManagerController(International)国际监管Claims Examiner Director of Operations运营总监Controller(General)Field Assurance Coordinator土地担保协调员GeneralManagerManagement Consultant管理顾问District ManagerHospital Administrator医院管理 PresidentImport/Export Manager进出口经理Product ManagerInsurance Claims Controller保险认领管理员Program ManagerInsurance Coordinator保险协调员Project Manager Inventory Control Manager库存管理经理Regional ManagerManager(Non-Profit and Charities)非盈利性慈善机构管理 Service ManagerManufacturing Manager制造业经理Vending ManagerTelecommunications Manager电信业经理Vice-PresidentTransportation Manager运输经理 Warehouse Manager 应聘职位名称（中英文对照）Administration(行政部分）Administrative Director 行政主管File Clerk Executive Assistant 行政助理 Office Manager Executive Secretary 行政秘书Receptionist General Office Clerk 办公室文员 Secretary Inventory Control Analyst 存货控制分析 Staff AssistantMail Room Supervisor 信件中心管理员 Stenographer 食品服务经理医疗保险管理操作经理生产经理房地产经理部门经理主考官管理员总经理市区经理总统产品经理程序管理经理项目经理区域经理服务经理售买经理副总裁仓库经理档案管理员办公室经理接待员秘书助理速记员Order Entry ClerkShipping/Receiving Expediter Vice-President of Administration订单输入文员收发督导员行政副总裁Telephone Operator Ticket Agent Typist电话操作员票务代理打字员应聘职位名称（中英文对照）Human Resources（人力资源部分）Assistant Personnel Officer Assistant Vice-President of Human ResourcesDirector of Human ResourcesJob Placement Officer Labor Relations Specialist Training Coordinator人事助理Benefits Coordinator员工福利协调员人力资源副总裁助理 Compensation Manager 薪酬经理人力资源总监人员配置专员劳动关系专员培训协调员Employment Consultant 招募顾问 Facility Manager Personnel Consultant Personnel Manager Recruiter Training Specialist 后勤经理员工顾问职员经理招聘人员培训专员Employer Relations Representative 员工关系代表Vice-President of Human Resources人力资源副总裁应聘职位名称（中英文对照）Marketing and Sales(市场与销售部分）Vice-President of SalesVice-President of MarketingSenior Account Manager Sales AdministratorRegional Sales ManagerRegional Account ManagerReal Estate AppraiserMerchandising ManagerMarketing ConsultantMarketing AssistantMarketing and Sales Director Market Research AnalystManufacturer's RepresentativeDirector of Subsidiary RightsCallback Representative销售副总裁市场副总裁销售主管Wholesale BuyerTravel Agent Telemarketer批发采购员旅行代办员电话销售员电话调查员销售员销售代表销售经理销售执行者销售助理零售采购员房地产经理房地产经纪人采购代理产品开发高级客户经理 Telemarketing Director电话销售总监地区销售经理 Tele-Interviewer地区客户经理 Salesperson 房地产评估师 SalesRepresentative 采购经理市场顾问市场助理Sales ManagerSales ExecutiveSales Assistant市场与销售总监Retail Buyer市场调查分析员Real Estate Manager厂家代表复查代表Real Estate Broker Product Developer分公司权利总监 Purchasing AgentAssistant Account Executive客户管理助理Marketing ManagerAdvertising Manager广告经理Marketing InternAdvertising Coordinator广告协调员Marketing DirectorAdvertising Assistant广告助理 Insurance Agent Ad Copywriter(Direct Mail)广告文撰写人 Account ManagerAccount Representative客户代表应聘职位名称（中英文对照）Accounting and Finance(会计与财务部分）Accounting Payable Clerk 应付帐款文员 Accounting Assistant Assistant Portfolio Manager 组合基金经理助理Accounting ManagerAccounts Receivable Clerk 应收帐款文员Accounting Clerk Certified Public Accountant 注册会计师Senior Accoutant Chief Financial Officer 首席财务官 Audit Manager Collections Officer 收款负责人Actuarial Anaylst Insurance Underwriter 保险承销商Auditor Bank Administrator 银行事务管理员 Junior Accountant Loan Administrator 贷款管理员 Bank Treasurer Management Accountant 管理会计 Billing Clerk Mortgage Underwriter 抵抻保险员Billing Supervisor Payroll Manager 工资经理Bookkeeper Staff Auditor 审计员 Bookkeeping Clerk Stock Broker 股票经纪人Budget Analyst T ax Accountant 税务会计Credit Analyst Tax Inspector税务检查员Credit ManagerVice-President of Administration and Finance财务行政副总裁 Financial Analyst Vice-President of Finance财务副总裁Financial Consultant Loan Servicer 贷款服务Financial Manager Financial Planner财务计划员Bank Clerk应聘职位名称（中英文对照）Engineering（工程部分）市场经理市场实习市场总监保险代理人客户经理会计助理会计经理会计文员高级会计审计经理保险分析员审计师初级会计资金调拨票据文员票据管理员档案管理档案管理助理预算分析信用分析信用管理经理财务分析财务顾问财务经理银行出纳Aerospace EngineerElectronics EngineerEngineering ConsultantEnvironmental EngineerFacilities Engineer Industrial EngineerManufacturing EngineerMechanical EngineerPetroleum Engineer航空工程师电子工程师工程顾问环境工程师设备工程师工业工程师机械工程师石油工程师Ceramic EngineerChemical EngineerCivil Engineer Electrical EngineerField EngineerMarine EngineerPlastics EngineerProduct Engineer陶器工程师化学工程师土木工程师电力工程师施工工程师航海工程师核能工程师整形工程师产品工程师制造业工程师 Nuclear EngineerA Useful Glossary for Educational Background(教育程度常用词汇）educationeducational background学历教育程度educational historycurriculum学历style='font-size: 12.0pt;font-family 第五篇：“紫色”词汇中英文对照“紫色” purple;violet 紫罗兰色 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radical form数学
Radical Form数学一、概述Radical Form是数学中的一个重要概念，它在代数运算和方程求解中发挥着重要作用。
本文将围绕radical form展开详细讨论，包括其定义、性质和应用。
二、radical form的定义1. radical form的概念在代数学中，radical form是指一个数或量的开平方的代数表达式，一般形式为√n，其中n为被开方数，且n为非负实数。
√9表示被开方数为9的根号表达式，其结果为3。
2. radical form的特点(1) radical form的指数必须为奇数在radical form中，被开方数的指数必须为奇数。
如果指数为偶数，那么开方的结果将是正数，而无法表达负数的情况。
radical form可以用根号符号表示，也可以用指数形式表示。
√9可以表示为9的1/2次方，即9^(1/2)，这两种表示形式是等价的。
三、radical form的运算1. radical form的加法和减法(1) 加法：对于两个radical form，其被开方数相同，可以直接进行加法。
√3+√3=2√3。
(2) 减法：同样地，对于两个radical form，其被开方数相同，可以直接进行减法。
√5-√2=√5-√2。
2. radical form的乘法和除法(1) 乘法：对于两个radical form，可以利用乘法的交换律和结合律进行合并。
√2*√3=√(2*3)=√6。
(2) 除法：在进行radical form的除法时，要先化简被开方数的因数，然后再进行开方。
√8/√2=√(8/2)=√4=2。
1. radical form在方程求解中的应用(1) 一次方程：对于一次方程，如果方程中含有radical form，可以利用开平方的性质将其转化为简单的算式，进而求解方程。
(2) 二次方程：在二次方程中，如果无法直接进行因式分解，那么可以利用radical form的概念，通过开方运算将方程转化为标准的二次方程求解。
美国高考sat数学试题及答案
美国高考sat数学试题及答案美国高考SAT数学试题及答案1. 某商店进行促销活动，所有商品打8折。
如果一件商品原价为$50，那么打折后的价格是多少？A. $40B. $45C. $35D. $20答案：B2. 一个长方形的长是宽的两倍，如果宽为$x$，那么长方形的周长是多少？A. $6x$B. $4x$C. $2x$D. $8x$答案：A3. 如果一个数的平方等于36，那么这个数是多少？A. 6B. -6C. 6 或 -6D. 0答案：C4. 在一次数学测试中，平均分是75分。
如果一个学生得了80分，那么他的分数比平均分高了多少？A. 5分B. 10分C. 15分D. 20分答案：A5. 一个圆的半径是5厘米，那么这个圆的面积是多少平方厘米？A. 25πB. 50πC. 75πD. 100π答案：B6. 如果一个函数$f(x) = 2x + 3$，那么$f(-1)$的值是多少？A. -1B. 1C. 5D. 7答案：B7. 一个等差数列的首项是3，公差是2，那么这个数列的第10项是多少？A. 23B. 21C. 19D. 17答案：A8. 如果一个三角形的两边长分别是5和7，且这两边夹角是90度，那么这个三角形的面积是多少？A. 12.5B. 15C. 17.5D. 20答案：A9. 如果一个函数$g(x) = x^2 - 4x + 3$，那么这个函数的最小值是多少？A. -1B. 0C. 1D. 3答案：A10. 在一个装有红球和蓝球的袋子里，红球和蓝球的比例是2:3。
如果随机抽取一个球，抽到红球的概率是多少？A. 2/5B. 3/5C. 4/5D. 1/2答案：A结束语：以上是美国高考SAT数学部分的试题及答案，希望对准备参加SAT考试的学生有所帮助。
SAT数学主要考点总结
SAT数学主要考点总结
SAT数学考点比较多，但是最重要的还是关于代数，算术，几何，数据分析和概述这几个方面的的。
大家在做SAT数学真题的时候会发现这些是出现比较多的，所以在备考的时候我们也要做到总结。
下面小编就为大家整理了SAT 数学主要考点总结的相关内容，供大家参考。
SAT代数重要考点(Algebra)
1. 数的乘方及开方
2. 变量和代数表达式
3. 因式分解
4. 方程
5. 代数不等式
6. 数列
7. 负指数与分数指数
8. 函数
9. 函数图像移动
SAT算术重要考点(Arithmetic)
1. 算术法则
2. 整数的概念和性质
3. 整数的正负性、奇偶性和质合性
4. 分数、小数和百分比
5. 因子和质因子
6. 连续整数及其性质
7. 数的开方和乘方
8. 数的除法和整除问题
9. 最大会约数和最小公倍数
10. 同余
11. 平均数/中值/众数
12. 数字推理
SAT几何重要考点(Geometry)
1. 平面几何
2. 立体几何
3. 平面直角坐标系
SAT数据分析和概率重要考点(Data Analysis and Probability)
1. 数据分析
2. 数据解释
3. 排列、组合和概率
4. 集合
以上就是小编为大家整理的关于SAT数学主要考点总结的相关内容，为大家总结了SAT数学真题中出现比较多的考点的总结，希望能够帮助大家更好的进行SAT数学备考，更多关于SAT数学的相关内容，请关注海知音教育官网。
新SAT数学题目中的重要公式和句型
新SAT 数学题目中的重要公式和句型－－北京新东方学校SAT 项目张志峰1. Triangle inequality / 三角形不等式三角形的两边之和，大于第三边2. 补角（supplementary angles）与余角（complementary angles）如果两个角互为补角（supplementary angles），那么它们的和是180 度如果两个角互为余角（complementary angles），那么它们的和是90 度3. 勾股定理（Pythagorean Theorem）4. Trigonometrysine，cosine，tangent 的正负，如图所示：5. 多边形的内角和如果一个多边形（polygon）的边的个数是n，它的内角和是：6. 多边形的外角和7. 正多边形的外角和性质公式如果一个正多边形（regular polygon）的边的个数是n，外角（exterior angle）是A，那么：8. 如何判断两条边平行？1）同位角（corresponding angles）相等，两条边平行2）内错角（alternate angles）相等，两条边平行3）同旁内角（interior angles on the same side）互补，两条边平行9. 柱体的体积计算公式柱体分两种，圆柱（cylinder）和棱柱（prism），10. 锥体的体积计算公式锥体分两种，圆锥（cone）和棱锥（pyramid），11. 扇形圆心角，弧长，扇形面积的万能公式注意，扇形的角a与圆的周角，扇形对应的弧长与圆的周长，扇形的面积与圆的面积的比值是一个常数12. 圆形的解析式13. 通过圆形的解析式求圆的半径如果圆形的解析式是标准式：，那么圆的半径是r如果圆形的解析式是展开式：，那么圆的半径是：14. 倍数关系twice as many A as B: A = 2Bhalf as many A as B: B = 2Athree times as many A as B: A = 3B15. 齿轮问题齿轮问题的常见考点是：16. complex numberDivisionExponentiationFormula:17.19. 图像对称（Reflection）点的对称：点（a, b）关于x-axis 对称：（a, - b）点（a, b）关于y-axis 对称：（- a, b）点（a, b）关于origin ／x-axis 和y-axis 对称：（- a, - b）点（a, b）关于y = x 对称：（b, a）函数图像的对称：函数y = f (x) 关于x-axis 对称：- f(x)函数y = f (x) 关于y-axis 对称：f(- x)函数y = f (x) 关于origin 对称：- f(- x)20. stretch（拉伸）和compress（压缩）stretch vertically by a factor of 2:compress vertically by a factor of 2:stretch horizontally by a factor of 2:compress horizontally by a factor of 2:21. 直线斜率求解方法1）通过两点坐标确定斜率如果A 点的坐标是, B 点的坐标是那么，过A 点和B 点的直线的斜率是：2）两条直线垂直（perpendicular），那么它们的斜率的乘积（product）= -1 3）两条直线平行（parallel），那么它们的斜率相等22. 直线方程的求解方法1）slope – intercept如果一条直线的slope 是k，y-intercept 是b，那么直线方程是：2）slope – point如果一条直线的slope 是k，经过点（a, b），那么直线方程是：23. 二次函数的standard formstandard form：1）a > 0，图像开口方向：upwarda < 0，图像开口方向：downward2）a 的绝对值变大，图像的开口的宽窄度：变窄a 的绝对值变小，图像的开口的宽窄度：变宽3）c：是二次函数的常数项，也是图像的y-intercept4）通过二次函数的standard form求图像的对称轴：5）通过二次函数的standard form 求图像的顶点坐标：6）求根公式：24. 二次函数的factored formfactored form：1）a > 0，图像开口方向：upwarda < 0，图像开口方向：downward2）a 的绝对值变大，图像的开口的宽窄度：变窄a 的绝对值变小，图像的开口的宽窄度：变宽3）r1和r2是二次函数的x-intercept4）通过二次函数的factored form 求图像的对称轴：5）通过二次函数的factored form 求图像的顶点坐标：25. 二次函数的vertex formvertex form：1）a > 0，图像开口方向：upwarda < 0，图像开口方向：downward2）顶点坐标：（p, q）26. 二次函数的判别式（discriminant）27. 韦达定理（Vieta`s Theorem）28. 多项式（polynomial）注意，多项式（polynomial）包括单项式，二项式，以及三项式1）等号两边，常数项相等，x 同次（degree）的系数（coefficient）相等2）多项式函数的常数项（constant）是函数图像与y 轴的截距（y-intercept）29. 多项式函数的余数定理（Polynomial Remainder Theorem）30. 指数运算规则1）2）3）。
SAT数学的三角形知识讲解一
SAT数学的三角形知识讲解一SAT数学考试中的三角形知识有哪些？SAT资料下载的小编为考生们整理了这些知识，让我们一起来学习一下吧！Triangles pop up all over the Math section. There are questions specifically about triangles, questions that ask about triangles inscribed in polygons and circles, and questions about triangles in coordinate geometry.Three Sides, Four Fundamental PropertiesEvery triangle, no matter how special, follows four main rules.1. Sum of the Interior AnglesIf you were trapped on a deserted island with tons of SAT questions about triangles, this is the one rule you’d need to know:The sum of the interior angles of a triangle is 180°.If you know the measures of two of a triangle’s angles, you’ll always be able to find the third by subtracting the sum of the first two from 180.2. Measure of an Exterior AngleThe exterior angle of a triangle is always supplementary to the interior angle with which it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex. In the image below, d is the exterior angle.Since d and c together form a straight angle, they are supplementary:. According to the first rule of triangles, the three angles of a triangle always add up to, so. Sinceand, d must equal a + b.3. Triangle Inequality RuleIf triangles got together to write a declaration of independence, they’d have a tough time, since one of their defining rules would be this:The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.There you have it: Triangles are unequal by definition.Take a look at the figure below:The triangle inequality rule says that c – b < a < c + b. The exact length of side a depends on the measure of the angle created by sides b andc. Witness this triangle:Using the triangle inequality rule, you can tell that 9 – 4 < x < 9 + 4, or 5 < x < 13. The exact value of x depends on the measure of the angle opposite side x. If this angle is large (close to) then x will be large (close to 13). If this angle is small (close to), then x will be small (close to 5).The triangle inequality rule means that if you know the length of two sides of any triangle, you will always know the range of possible side lengths for the third side. On some SAT triangle questions, that’s all you’ll need.4. Proportionality of TrianglesHere’s the final fundamental triangle property. This one exp lains the relationships between the angles of a triangle and the lengths of the triangle’s sides.In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.In this figure, side a is clearly the longest side andis the largest angle. Meanwhile, side c is the shortest side andis the smallest angle. So c < b < a and C < B < A. This proportionality of side lengths and angle measures holds true for all triangles.See if you can use this rule to solve the question below:What is one possible value of x if angle C < A < p>(A)1(B)6(C)7(D)10(E)15According to the proportionality of triangles rule, the longest side of a triangle is opposite the largest angle. Likewise, the shortest side of a triangle is opposite the smallest angle. The largest angle in triangle ABC is, which is opposite the side of length 8. The smallest angle in triangleABC is, which is opposite the side of length 6. This means that the third side, of length x, measures between 6 and 8 units in length. The only choice that fits the criteria is 7. C is the correct answer.Special TrianglesSpecial triangles are “special” not because they get to follow fewer rules than other triangles but because they get to follow more. Each type of special triangle has its own special name: isosceles, equilateral, and right. Knowing the properties of each will help you tremendously, humongously, a lot, on the SAT.But first we have to take a second to explain the markings we use to describe the properties of special triangles. The little arcs and ticks drawn in the figure below show that this triangle has two sides of equal length and three equal angle pairs. The sides that each have one tick through them are equal, as are the sides that each have two ticks through them. The angles with one little arc are equal to each other, the angles with two little arcs are equal to each other, and the angles with three little arcs are all equal to each other.Isosceles TrianglesIn ancient Greece, Isosceles was the god of triangles. His legs were of perfectly equal length and formed two opposing congruent angles when he stood up straight. Isosceles triangles share many of the same properties, naturally. An isosceles triangle has two sides of equal length, and those two sides are opposite congruent angles. These equal angles are usually called as base angles. In the isosceles triangle below, side a = b and:If you know the value of one of the base angles in an isosceles triangle, you can figure out all the angles. Let’s say you’ve got an isosceles triangle with a base angle of 35º. Since you know isosceles triangles have two congruent base angles by definition, you know that the other base angle is also 35º. All three angles in a triangle must always add up to 180º, right? Correct. Tha t means you can also figure out the value of the third angle: 180º –35º –35º = 110º.Equilateral TrianglesAn equilateral triangle has three equal sides and three congruent 60ºangles.Based on the proportionality rule, if a triangle has three equal sides, that triangle must also have three equal angles. Similarly, if you know that a triangle has three equal angles, then you know it also has three equal sides.Right TrianglesA triangle that contains a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. The angles opposite the legs of a right triangle are pleme ntary (they add up to 90º).In the figure above,is the right angle (as indicated by the box drawn in the angle), side c is the hypotenuse, and sides a and b are the legs.If triangles are an SAT favorite, then right triangles are SAT darlings. In other words, know these rules. And know the Pythagorean theorem.The Pythagorean TheoremThe Greeks spent a lot of time reading, eating grapes, and riding around on donkeys. They also enjoyed the occasional mathematical epiphany. One day, Pythagoras discovered that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. “Eureka!” he said, and the SAT had a new topic to test.Here’s the Pythagorean theorem: In a right triangle, a2 + b2 = c2:where c is the length of the hypotenuse and a and b are the lengths of the two legs.The Pythagorean theorem means that if you know the measures of two sides of a right triangle, you can always find the third. “Eureka!” you say.Pythagorean TriplesBecause right triangles obey the Pythagorean theorem, only a specific few have side lengths that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse of length= 5.83.The few sets of three integers that do obey the Pythagorean theorem and can therefore be the lengths of the sides of a right triangle are called Pythagorean triples. Here are some mon ones:{3, 4, 5}{5, 12, 13}{7, 24, 25}{8, 15, 17}In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}.The SAT is full of right triangles whose side lengths are Pythagorean triples. Study the ones above and their multiples. Identifying Pythagorean triples will help you cut the amount of time you spend doing calculations. In fact, you may not have to do any calculations if you get these down cold.Extra-Special Right TrianglesRight triangles are pretty special in their own right. But there are twoextra-special right triangles. They are 30-60-90 triangles and 45-45-90triangles, and they appear all the time on the SAT.In fact, knowing the rules of these two special triangles will open up all sorts of time-saving possibilities for you on the test. Very, very often, instead of having to work out the Pythagorean theorem, you’ll be able to apply the standard side ratios of either of these two types of triangles, cutting out all the time you need to spend calculating.。
对数、三角函数、解三角形
23. If sinx cos 2x and 0 x 2 , then find x. (MAθ 1992)
24. Prove the following equalities. (M&IQ 1992) i. sin 10°sin 20°sin 30°= sin 10°sin 10°sin 100°； ii. sin 20°sin 20°sin 30°= sin 10°sin 20°sin 80°； iii. sin 20°sin 30°sin 30°= sin 10°sin 40°sin 50°.
38.In ADC , angle bisector DB is drawn. If AB 3, AD 6, and CD = 8, find BD. (MAθ 1987)
39. If the sides of a triangle are in the ratio 4:6:8, then find the cosine of the smallest angle. (MAθ1991)
一个推论ABC ab sin C abc , .
2
4R
EXAMPLE 3-8 Find the circumradius of a triangle whose sides are 13,14, and 15.
EXERCISE 3-6 Show that [ABC] = 2R2 sin A sin B sin C.
a b2 c2 c2 a b2 16
a b ca b ca b c a b c 16
SAT数学多边形知识讲解下载
SAT数学多边形知识讲解下载SAT数学考试中的多边形知识有哪些？SAT资料下载的小编为考生们整理了这些知识，让我们一起来学习一下吧！A polygon is a two-dimensional figure with three or more straight sides. (So triangles are actually a type of polygon.) Polygons are named according to the number of sides they have.All polygons, no matter how many sides they possess, share certain characteristics:The sum of the interior angles of a polygon with n sides is (n – 2). For instance, the sum of the interior angles of an octagon is (8 – 2)= 6=.The sum of the exterior angles of any polygon is.The perimeter of a polygon is the sum of the lengths of its sides. The perimeter of the hexagon below is 5 + 4 + 3 + 8 + 6 + 9 = 35.Regular PolygonsThe polygon whose perimeter you just calculated was an irregular polygon. But most of the polygons on the SAT are regular: Their sides are of equal length and their angles congruent. Neither of these conditions can exist without the other. If the sides are all equal, the angles will all be congruent, and vice versa. In the diagram below, you’ll see, from left to right, a regular pentagon, a regular octagon, and a square (also known as a regular quadrilateral):QuadrilateralsGood news: Most polygons on the SAT have just four sides. You won’t have to tangle with any dodecahedrons on the SAT you take. But this silver cloud actually has a dark lining: There are five different types of quadrilaterals that pop up on the test. These five quadrilaterals are trapezoids, parallelograms, rectangles, rhombuses, and squares.TrapezoidsA trapezoid may sound like a new Star Wars character. Certainly, it would be less annoying than Jar Jar Binks. But it’s actually the name of a quadrilateral with one pair of parallel sides and one pair of nonparallel sides.In this trapezoid, AB is parallel to CD (shown by the arrow marks), whereasAC and BD are not parallel.The formula for the area of a trapezoid iswhere s1 and s2 are the lengths of the parallel sides (also called the bases of the trapezoid), and h is the height. In a trapezoid, the height is the perpendicular distance from one base to the other.To find the area of a trapezoid on the SAT, you’ll often have to use your knowledge of triangles. Try to find the area of the trapezoid pictured below:The question tells you the length of the bases of this trapezoid, 6 and 10. But to find the area, you first need to find the height. To do that, split the trapezoid into a rectangle and a 45-45-90 triangle by drawing in the height.Once, you’ve drawn in the height, you can split the base that’s equal to 10into two parts: The base of the rectangle is 6, and the leg of the triangle is 4. Since the triangle is 45-45-90, the two legs must be equal. This leg, though, is also the height of the trapezoid. So the heightof the trapezoid is 4. Now you can plug the numbers into the formula:ParallelogramA parallelogram is a quadrilateral whose opposite sides are parallel.In a parallelogram,Opposite sides are equal in length: BC = AD and AB = DCOpposite angles are equal:andAdjacent angles are supplementary:The diagonals bisect (split) each other: BE = ED and AE = ECOne diagonal splits a parallelogram into two congruent triangles:Two diagonals split a parallelogram into two pairs of congruent triangles:andThe area of a parallelogram is given by the formulawhere b is the length of the base, and h is the height.RectanglesA rectangle is a quadrilateral in which the opposite sides are parallel and the interior angles are all right angles. Another way to look at rectangles is as parallelograms in which the angles are all right angles. As with parallelograms, the opposite sides of a rectangle are equal.The formula for the area of a rectangle iswhere b is the length of the base, and h is the height.The diagonals of a rectangle are always equal to each other. And one diagonal through the rectangle cuts the rectangle into two equal right triangles. In the figure below, the diagonal BD cuts rectangle ABCD into congruent right triangles ABD and BCD.Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, if you know two of these values, you can always calculate the third with the Pythagorean theorem. If you know the side lengths of the rectangle, you can calculate the diagonal. If you know the diagonal and one side length, you can calculate the other side. Also, keep in mind that the diagonal might cut the rectangle into a 30-60-90triangle. That would make your calculating job even easier.RhombusA rhombus is a specialized parallelogram in which all four sides are of equal length.In a rhombus,All four sides are equal: AD = DC = CB = BAThe diagonals bisect each other and form perpendicular lines (but note that the diagonals are not equal in length)The diagonals bisect the vertex anglesThe formula for the area of a rhombus iswhere b is the length of the base and h is the height.To find the area of a rhombus on the SAT (you guessed it), you’ll probably have to split it into triangles:If ABCD is a rhombus, AC = 4, and ABD is an equilateral triangle, what is the area of the rhombus?Since ABD is an equilateral triangle, the length of each side of the rhombus must be 4, and angles ADB and ABD are 60º. All you have to do is find the height of the rhomb us. Draw an altitude from A to DC to create a 30-60-90triangle.Since the hypotenuse of the 30-60-90 triangle is 4, you can use the ratio 1::2 to calculate that the length of this altitude is 2. The area formula for a rhombus is bh, so the area of this rhombus is 42= 8.SquareA square combines the special features of the rectangle and rhombus: All its angles are 90º, and all four of its sides are equal in length.The square has two more crucial special qualities. In a square,Diagonals bisect each other at right angles and are equal in length.Diagonals bisect the vertex angles to create 45º angles. (Th is means that one diagonal will cut the square into two 45-45-90 triangles, while two diagonals break the square into four 45-45-90 triangles.)The formula for the area of a square iswhere s is the length of a side of the square.Because a diagonal drawn into the square forms two congruent 45-45-90triangles, if you know the length of one side of the square, you can always calculate the length of the diagonal:Since d is the hypotenuse of the 45-45-90 triangle that has legs of length 5, according to the ratio 1:1:, you know that.Similarly, if you know the length of the diagonal, you can calculate the length of the sides of the square.。
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江西省南昌市2015-2016学年度第一学期期末试卷（江西师大附中使用）高三理科数学分析一、整体解读试卷紧扣教材和考试说明，从考生熟悉的基础知识入手，多角度、多层次地考查了学生的数学理性思维能力及对数学本质的理解能力，立足基础，先易后难，难易适中，强调应用，不偏不怪，达到了“考基础、考能力、考素质”的目标。
试卷所涉及的知识内容都在考试大纲的范围内，几乎覆盖了高中所学知识的全部重要内容，体现了“重点知识重点考查”的原则。
1．回归教材，注重基础试卷遵循了考查基础知识为主体的原则，尤其是考试说明中的大部分知识点均有涉及，其中应用题与抗战胜利70周年为背景，把爱国主义教育渗透到试题当中，使学生感受到了数学的育才价值，所有这些题目的设计都回归教材和中学教学实际，操作性强。
2．适当设置题目难度与区分度选择题第12题和填空题第16题以及解答题的第21题，都是综合性问题，难度较大，学生不仅要有较强的分析问题和解决问题的能力，以及扎实深厚的数学基本功，而且还要掌握必须的数学思想与方法，否则在有限的时间内，很难完成。
3．布局合理，考查全面，着重数学方法和数学思想的考察在选择题，填空题，解答题和三选一问题中，试卷均对高中数学中的重点内容进行了反复考查。
包括函数，三角函数，数列、立体几何、概率统计、解析几何、导数等几大版块问题。
这些问题都是以知识为载体，立意于能力，让数学思想方法和数学思维方式贯穿于整个试题的解答过程之中。
二、亮点试题分析1．【试卷原题】11.已知,,A B C 是单位圆上互不相同的三点，且满足AB AC →→=，则AB AC →→⋅的最小值为（ ）A ．14-B ．12-C ．34-D ．1-【考查方向】本题主要考查了平面向量的线性运算及向量的数量积等知识，是向量与三角的典型综合题。
解法较多，属于较难题，得分率较低。
【易错点】1．不能正确用OA ，OB ，OC 表示其它向量。
2．找不出OB 与OA 的夹角和OB 与OC 的夹角的倍数关系。
【解题思路】1．把向量用OA ，OB ，OC 表示出来。
2．把求最值问题转化为三角函数的最值求解。
【解析】设单位圆的圆心为O ，由AB AC →→=得，22()()OB OA OC OA -=-，因为1OA OB OC ===，所以有，OB OA OC OA ⋅=⋅则()()AB AC OB OA OC OA ⋅=-⋅-2OB OC OB OA OA OC OA =⋅-⋅-⋅+ 21OB OC OB OA =⋅-⋅+设OB 与OA 的夹角为α，则OB 与OC 的夹角为2α所以，cos 22cos 1AB AC αα⋅=-+2112(cos )22α=--即，AB AC ⋅的最小值为12-，故选B 。
【举一反三】【相似较难试题】【2015高考天津，理14】在等腰梯形ABCD 中,已知//,2,1,60AB DC AB BC ABC ==∠= ,动点E 和F 分别在线段BC 和DC 上,且,1,,9BE BC DF DC λλ==则AE AF ⋅的最小值为 .【试题分析】本题主要考查向量的几何运算、向量的数量积与基本不等式.运用向量的几何运算求,AE AF ，体现了数形结合的基本思想，再运用向量数量积的定义计算AE AF ⋅，体现了数学定义的运用，再利用基本不等式求最小值，体现了数学知识的综合应用能力.是思维能力与计算能力的综合体现. 【答案】2918【解析】因为1,9DF DC λ=12DC AB =，119199918CF DF DC DC DC DC AB λλλλλ--=-=-==， AE AB BE AB BC λ=+=+，19191818AF AB BC CF AB BC AB AB BC λλλλ-+=++=++=+，()221919191181818AE AF AB BC AB BC AB BC AB BCλλλλλλλλλ+++⎛⎫⎛⎫⋅=+⋅+=+++⋅⋅ ⎪ ⎪⎝⎭⎝⎭19199421cos1201818λλλλ++=⨯++⨯⨯⨯︒2117172992181818λλ=++≥+= 当且仅当2192λλ=即23λ=时AE AF ⋅的最小值为2918. 2．【试卷原题】20. （本小题满分12分）已知抛物线C 的焦点()1,0F ，其准线与x 轴的交点为K ，过点K 的直线l 与C 交于,A B 两点，点A 关于x 轴的对称点为D ． （Ⅰ）证明：点F 在直线BD 上； （Ⅱ）设89FA FB →→⋅=，求BDK ∆内切圆M 的方程. 【考查方向】本题主要考查抛物线的标准方程和性质，直线与抛物线的位置关系，圆的标准方程，韦达定理，点到直线距离公式等知识，考查了解析几何设而不求和化归与转化的数学思想方法，是直线与圆锥曲线的综合问题，属于较难题。
【易错点】1．设直线l 的方程为(1)y m x =+，致使解法不严密。
2．不能正确运用韦达定理，设而不求，使得运算繁琐，最后得不到正确答案。
【解题思路】1．设出点的坐标，列出方程。
2．利用韦达定理，设而不求，简化运算过程。
3．根据圆的性质，巧用点到直线的距离公式求解。
【解析】（Ⅰ）由题可知()1,0K -，抛物线的方程为24y x =则可设直线l 的方程为1x my =-，()()()112211,,,,,A x y B x y D x y -，故214x my y x =-⎧⎨=⎩整理得2440y my -+=，故121244y y m y y +=⎧⎨=⎩则直线BD 的方程为()212221y y y y x x x x +-=--即2222144y y y x y y ⎛⎫-=- ⎪-⎝⎭令0y =，得1214y yx ==，所以()1,0F 在直线BD 上.（Ⅱ）由（Ⅰ）可知121244y y m y y +=⎧⎨=⎩，所以()()212121142x x my my m +=-+-=-，()()1211111x x my my =--= 又()111,FA x y →=-，()221,FB x y →=-故()()()21212121211584FA FB x x y y x x x x m →→⋅=--+=-++=-，则28484,93m m -=∴=±，故直线l 的方程为3430x y ++=或3430x y -+=213y y -===±，故直线BD 的方程330x -=或330x -=，又KF 为BKD ∠的平分线，故可设圆心()(),011M t t -<<，(),0M t 到直线l 及BD 的距离分别为3131,54t t +--------------10分 由313154t t +-=得19t =或9t =（舍去）.故圆M 的半径为31253t r +== 所以圆M 的方程为221499x y ⎛⎫-+= ⎪⎝⎭【举一反三】【相似较难试题】【2014高考全国，22】 已知抛物线C ：y 2＝2px(p>0)的焦点为F ，直线y ＝4与y 轴的交点为P ，与C 的交点为Q ，且|QF|＝54|PQ|.（1）求C 的方程；（2）过F 的直线l 与C 相交于A ，B 两点，若AB 的垂直平分线l′与C 相交于M ，N 两点，且A ，M ，B ，N 四点在同一圆上，求l 的方程．【试题分析】本题主要考查求抛物线的标准方程,直线和圆锥曲线的位置关系的应用,韦达定理,弦长公式的应用,解法及所涉及的知识和上题基本相同. 【答案】（1）y 2＝4x. （2）x －y －1＝0或x ＋y －1＝0. 【解析】（1）设Q(x 0，4)，代入y 2＝2px ，得x 0＝8p，所以|PQ|＝8p ，|QF|＝p 2＋x 0＝p 2＋8p.由题设得p 2＋8p ＝54×8p ，解得p ＝－2(舍去)或p ＝2，所以C 的方程为y 2＝4x.（2）依题意知l 与坐标轴不垂直，故可设l 的方程为x ＝my ＋1(m≠0)． 代入y 2＝4x ，得y 2－4my －4＝0. 设A(x 1，y 1)，B(x 2，y 2)， 则y 1＋y 2＝4m ，y 1y 2＝－4.故线段的AB 的中点为D(2m 2＋1，2m)， |AB|＝m 2＋1|y 1－y 2|＝4(m 2＋1)．又直线l ′的斜率为－m ，所以l ′的方程为x ＝－1m y ＋2m 2＋3.将上式代入y 2＝4x ，并整理得y 2＋4m y －4(2m 2＋3)＝0.设M(x 3，y 3)，N(x 4，y 4)，则y 3＋y 4＝－4m，y 3y 4＝－4(2m 2＋3)．故线段MN 的中点为E ⎝ ⎛⎭⎪⎫2m2＋2m 2＋3，－2m ，|MN|＝1＋1m 2|y 3－y 4|＝4（m 2＋1）2m 2＋1m 2.由于线段MN 垂直平分线段AB ，故A ，M ，B ，N 四点在同一圆上等价于|AE|＝|BE|＝12|MN|，从而14|AB|2＋|DE|2＝14|MN|2，即 4(m 2＋1)2＋⎝ ⎛⎭⎪⎫2m ＋2m 2＋⎝ ⎛⎭⎪⎫2m 2＋22＝4（m 2＋1）2（2m 2＋1）m 4，化简得m 2－1＝0，解得m ＝1或m ＝－1， 故所求直线l 的方程为x －y －1＝0或x ＋y －1＝0.三、考卷比较本试卷新课标全国卷Ⅰ相比较，基本相似，具体表现在以下方面： 1. 对学生的考查要求上完全一致。
即在考查基础知识的同时，注重考查能力的原则，确立以能力立意命题的指导思想，将知识、能力和素质融为一体，全面检测考生的数学素养，既考查了考生对中学数学的基础知识、基本技能的掌握程度，又考查了对数学思想方法和数学本质的理解水平，符合考试大纲所提倡的“高考应有较高的信度、效度、必要的区分度和适当的难度”的原则． 2. 试题结构形式大体相同，即选择题12个，每题5分，填空题4 个，每题5分，解答题8个（必做题5个），其中第22，23，24题是三选一题。
题型分值完全一样。
选择题、填空题考查了复数、三角函数、简易逻辑、概率、解析几何、向量、框图、二项式定理、线性规划等知识点，大部分属于常规题型，是学生在平时训练中常见的类型．解答题中仍涵盖了数列，三角函数，立体何，解析几何，导数等重点内容。
3. 在考查范围上略有不同，如本试卷第3题，是一个积分题，尽管简单，但全国卷已经不考查了。
四、本考试卷考点分析表（考点/知识点，难易程度、分值、解题方式、易错点、是否区分度题）。