Mathematics

mathematics的意思用法总结mathematics有数学,〈诗〉同“ripen”,算学的意思。

那你们想知道mathematics的用法吗?今日我给大家带来了mathematics的用法,盼望能够关心到大家，一起来学习吧。

mathematics的意思n. 数学,〈诗〉同“ripen”,算学变形：形容词：mathematical; 副词：mathematically;mathematics用法mathematics可以用作名词mathematics的意思是“数学”,是讨论数字、数量和外形的科学,包括“算术、代数、几何”等学科。

其前一般不用冠词。

mathematics单复数同形,但指作为一个科学分支“数学”或作为一个学科的“数学”时,句中谓语动词要用单数形式; 假如是用来指这门学科中的详细内容或数学实践力量时,如“数学力量”“数学成果”“数学应用”“计算力量”“运算结果”等,则动词要用复数形式。

在英式口语中mathematics可写作maths;在美式口语中mathematics可写作math。

mathematics用作名词的用法例句Mathematics is her favorite subject.数学是她喜爱的科目。

He has a great faculty for mathematics.他具有很强的学数学的才能。

mathematics用法例句1、The degree provides a thorough grounding in both mathematics and statistics.该学位课程将为数学和统计学打下扎实的基础。

2、One in five young adults was struggling with everyday mathematics.1/5的年轻人做日常的数学计算都费劲。

3、We had a very good mathematics mistress who pulled me up.我们有个很精彩的女数学老师，她帮我提高了水平。

数学mathematics, maths（BrE）,math（AmE）被除数dividend除数divisor商quotient等于equals, is equal to,is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses（pl.）命题proposition算术arithmetic加plus（prep.）, add（v.）, addition（n.）被加数augend, summand加数addend和sum减minus（prep.）, subtract（v.）, subtraction（n.）被减数minuend减数subtrahend差remainder乘times（prep.）, multiply（v.）,multiplication（n.）被乘数multiplicand, faciend乘数multiplicator积product除divided by（prep.）, divide（v.）, division（n.）整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae（pl.）单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor,z-factor等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial 对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four,the fourth powern次方the power of n, thenth power开方evolution,extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four,the fourth rootn次方根the root of n,the nth root集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition 值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列,级数series微积分calculus微分differential导数derivative极限limit无穷大infinite（a.）infinity（n.）无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix 行列式determinant几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle 直角right angle钝角obtuse angle 平角straight angle 周角perigon底base边side高height三角形triangle锐角三角形acute triangle 直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem 钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus,rhombi（pl.）, diamond 正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon 十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon 圆circle圆心centre（BrE）, center（AmE）半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter面积area轨迹locus, loca（pl.）相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron 十二面体dodecahedron二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism 旋转rotation轴axis圆锥cone圆柱cylinder圆台frustum of a cone 球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space 坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incentre（BrE）,incenter（AmE）外心excentre（BrE）,excenter（AmE）旁心escentre（BrE）,escenter（AmE）垂心orthocentre（BrE）,orthocenter（AmE）重心barycentre（BrE）,barycenter（AmE）内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average 方差variance标准差root-mean-squaredeviation, standarddeviation比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution 正态分布normal distribution 非正态分布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagram。

Mathematics数学-附答案（1）MathematicsAnswer all questions1.Given that 2x+y= 16 and In (2x–y) = ? In 5, find the value of x and of y, leaving youranswers in surd form. [8]X=(4+√5)/3 Y=(8-√5)/32.Given that A = {x:-5 < 2x-3 ≤ 15}, B = {x: -4 ≤ x + 1 < 7} andA UB = {x: a ≤ x -1≤ b}, find the value of a and of b. [5]a=-6 b=83.Functions f, g and h are defined byf:x→ 2 + xg:x→ -|x-1|h:x→ -x2 + 2(a)Find the values of gh(-2) and f-1gh(-2).-3and-1 [4](b)Find in similar form, gf, fg and gh. gf=-|x+1| fg=2-|x-1| gh=-|-x2+1|[3](c)Find the values of x such that fg(x) = -1 x=4 or x=-2[3](d)Show that hg(x) = 1 + 2x- x2. ∵hg(x)=h(-|x-1|)=-(-|x-1|)2+2=-(x-1)2+2=-x2+2x+1 [4]4.Three points have coordinates A(5, -3), B(-2,1) and C(a , 5).(a)Find, in terms of a, the coordinates of M, which is the midpoint of BC. [4]The midpoint of BC is ((a-2)/2,3)(b)Find the value of a for which AM is perpendicular to ABa=132/7. [6]5.Find the fifth and sixth terms of binomial expansion of ( 2- ? x2)9. Hence find thecoefficient of x10 in ( 2- ? x2 )9 (3x2+5 ) . 5th:(-19/3)9 6th:( -10) 9[8]6.Find the coordinates of the points of intersection of the line x + y = 3 and the curvex2– 2x +2y2= 3. [6] x=3, y=0 or x=3/5,y=12/57.Find the range of values of k for which the equation x2– 6x + k2– 7 = 0 has real roots. 36-4×1×（k2-7）≥0 the result is -4≤k≤4 [6] 8.Find all the angles between 0° and 360° inclusive which satisfysinxsec2x– 2tan x = 0 x=60°[6] 9. A body moves in a straight line so that its displacement, S m, from a point O at time tsec, is given by S = 5 + 15t2– 5t3.Find(a)The time when the body is instantaneously at rest, [4](b)The acceleration when t = 4, Answer:75[3](c)The total distance moved by the body in 9 seconds, [6](d)The average speed of the moving body. [4]10.A semicircle of centre O and radius 10 cm has diameter AB. The chord AQ is10 cm and the ratio of arc AP to arc AQ is 2 : 3.(a)Show that ∠AOP is 2?9 π radians.∵AO=QO=AQ=10 cm ∴∠AOQ=60° and∵the ratio of arc AP to arc AQ is2 : 3. ∴∠AOP=40°(40°/360°)×2л=2л/9[6](b)Calculate the area of segment AQP. the area of segment AQP is (50л/3)-25√3 [4]11.If x2– 2x– 3 is a factor of the expression x4+ px3+ qx– 81, find the value of p and of q. With these of p and q, factorize the expression completely. [10] 数学回答以下所有问题1.已知2x+y和ln（2x-y）= ? In 5 求x和y的值，保留根号。

mathematics的意思用法总结mathematics有数学,〈诗〉同“ripen”,算学的意思。

那你们想知道mathematics的用法吗?今天给大家带来了mathematics的用法,希望能够帮助到大家，一起来学习吧。

mathematics的意思n. 数学,〈诗〉同“ripen”,算学变形：形容词：mathematical; 副词：mathematically;mathematics用法mathematics可以用作名词mathematics的意思是“数学”,是研究数字、数量和形状的科学,包括“算术、代数、几何”等学科。

其前一般不用冠词。

mathematics单复数同形,但指作为一个科学分支“数学”或作为一个学科的“数学”时,句中谓语动词要用单数形式; 如果是用来指这门学科中的具体内容或数学实践能力时,如“数学能力”“数学成绩”“数学应用”“计算能力”“运算结果”等,则动词要用复数形式。

在英式口语中mathematics可写作maths;在美式口语中mathematics可写作math。

mathematics用作名词的用法例句Mathematics is her favorite subject.数学是她喜欢的科目。

He has a great faculty for mathematics.他具有很强的学数学的才能。

mathematics用法例句1、The degree provides a thorough grounding in both mathematics and statistics.该学位课程将为数学和统计学打下扎实的基础。

2、One in five young adults was struggling with everyday mathematics.1/5的年轻人做日常的数学计算都费劲。

3、We had a very good mathematics mistress who pulled me up.我们有个很出色的女数学老师，她帮我提高了水平。

关于数学的英文单词一、数学的英文单词的拼写和发音数学的英文单词mathematics是一个复数名词，但它的意思是单数的，表示一门学科或一种科学。

因此，它的谓语动词要用单数形式，例如：Mathematics is my favorite subject.数学是我最喜爱的学科。

数学的英文单词mathematics的缩写有两种，一种是math，一种是maths。

它们的意思和用法都一样，只是在不同的地区有不同的习惯。

一般来说，美国和加拿大的人更喜欢用math，而英国、澳大利亚和其他英语国家的人更喜欢用maths。

例如：I'm good at math.我数学很好。

I'm good at maths.我数学很好。

二、数学的英文单词的用法数学的英文单词mathematics可以用来表示数学这门学科，也可以用来表示数学的内容或能力。

当表示数学这门学科时，它的前面一般不用冠词，当表示数学的内容或能力时，它的前面一般要用冠词。

例如：I like mathematics. 我喜欢数学（这门学科）。

I like the mathematics of fractals. 我喜欢分形的数学（内容）。

He has a talent for mathematics. 他有数学（能力）的天赋。

数学的英文单词mathematics可以和其他的词组合，形成一些专门的术语，表示数学的不同分支或领域。

例如：applied mathematics 应用数学pure mathematics 纯数学discrete mathematics 离散数学computational mathematics 计算数学mathematical logic 数理逻辑mathematical physics 数学物理mathematical biology 数学生物学数学的英文单词math或maths可以用来修饰其他的名词，表示与数学相关的事物或人。

统计数据一实验目的学习用Mathematica求来自某个总体的一个样本的样本均值，中位数, 样本方差，偏度, 峰度, 样本分位数和其他数字特征, 并能由样本作出直方图.二学习Mathematica命令1.调用统计软件包命令<<Statistics\ 和<<Statistics\用或进行统计数据的处理, 必须调用相应的软件包. 首先要输入并执行命令<<Statistics\<<Statistics\对版本，则也可以简单地输入并执行命令<<Statistics`便完成了数据概括的准备工作.2.调用作图软件包命令<<Graphics\用或作直方图, 必须调用相应的作图软件包. 输入并执行<<Graphics\对版本，则也可以简单地输入并执行命令<<Graphics`这时可以查询这个软件包中的一些作图命令的用法. 如输入BarChart则应该得到命令BarChart的用法说明(英文), 如果没有, 则说明调用软件包不成功, 必须重新启动计算机, 再次调用软件包.3.求样本数字特征命令（1）命令Mean[list], 给出样本list的均值;（2）命令Median[list] , 给出样本list的中位数;（3）命令Min[list] , 给出样本list的最小值;（4）命令Max[list] , 给出样本list的最大值;（5）命令Variance[list] , 给出样本list的方差;（6）命令StandardDeviation[list], 给出样本list的标准差;（7）命令Quantile[list,α], 给出样本list的α分位数;（8）命令CentralMoment[list, n], 给出样本list的n阶中心矩.4.计算分组后各组内含有的数据个数命令BinCounts命令BinCounts的格式是BinCounts[数据, {最小值, 最大值, 增量}]例如输入BinCounts[{1,1,2,3,4,4,5,15,6,7,8,8,8,9,10,13},{0,15,3}]输出为{4,4,5,1,2}上述输出表示指落入区间]3,0(, ]6,3(, ]9,6(, ]12,9(, ]15,12(的数据个数分别是4,4,5,1,2. 注意每个小区间是左开右闭的区间.5.作条形图命令BarChart命令BarChart 的格式是BarChart[数据, 选项1, 选项2, …]其中数据是}},,{},,{{2211 x y x y 或},,{21 y y 的形式. 而 ,,21y y 为条形的高度, ,,21x x 为条形的中心. 在数据为},,{21 y y 的形式时默认条形的中心是{1,2,…}. 常用选项有BarSpacing->数值1, BarGroupSpacing->数值2. 例如输入BarChart[{{4,},{4,},{5,},{1,},{2,}}, BarGroupSpacing->]则输出下面的条形图(图:图三 实验内容1. 样本的位置统计, 分散性统计, 样本中心矩, 分布的形状统计, 数据的变换.例1 某厂生产的某种型号的细轴中任取20个，测得其直径数据如下：, , , , , , , , , , , , , , , , , , ,求以上数据的样本均值, 中位数, 四分位数; 样本方差, 样本标准差, 极差, 变异系数, 二阶、三阶和四阶中心矩; 求偏度, 峰度. 并把数据中心化和标准化.解 首先输入并执行调用统计软件包命令<<Statistics`输入data1={, , , , , , , , , , , , , , , , , , , };(*数据集记为data1*)Mean[data1](*求样本均值*)Median[data1](*求样本中位数*)Quartile[data1](*求样本的分位数, 中位数, 及分位数*)Quantile[data1,](*求样本的分位数*)Quantile[data1,](*求样本的分位数*)执行后得到输出{, , }因此, 样本均值为, 样本中位数为, 样本的分位数为, 分位数是, 样本的分位数是, 样本的分位数是.输入 Variance[data1] (*求样本方差*)StandardDeviation[data1] (*求样本标准差*)VarianceMLE[data1] (*求样本方差2*)StandardDeviationMLE[data1] (*求样本标准差2*)SampleRange[data1] (*求样本极差*)输出为因此样本方差S 2=, 注意Variance 给出的是无偏估计时的方差. 其计算公式是∑=--=n i i x xn S 122)(11, 样本标准差S =)(2S S =. 而VarianceMLE 给出的是总体方差的极大似然估计, 它的计算公式是∑=-=n i i x xn S 122*)(1, 这里=2*S, 要比S 2略小. StandardDeviationMLE 给出的是总体标准差的极大似然估计: S *=. SampleRange 给出的是样本极差(样本极大减去样本极小), 这里极差R=.输入CoefficientOfVariation[data1](*求变异系数. 变异系数的定义是样本标准差与样本均值之比*)输出为.输入CentralMoment[data1,2](*求样本二阶中心矩*)CentralMoment[data1,3](*求样本三阶中心矩*)CentralMoment[data1,4](*求样本四阶中心矩*)输出为输入Skewness[data1](*求偏度, 偏度的定义是三阶中心矩除以标准差的立方*)Kurtosis[data1](*求峰度, 峰度的定义是四阶中心矩除以方差的平方*)输出结果为以上结果表明: 数据data1的偏度(Skewness)是, 而负的偏度表明总体分布密度有较长的右尾, 即分布向左偏斜. 数据(data1)的峰度( Kurtosis)为. 峰度大于3时表明总体的分布密度比有相同方差的正态分布的密度更尖锐和有更重的尾部. 峰度小于3时表明总体的分布密度比正态分布的密度更平坦或者有更粗的腰部.输入ZeroMean[data1](*把数据中心化, 即每个数据减去均值*)输出为{, , , , , ,, , , , , , ,, , , , , , }输入Standardize[data1](*把数据标准化, 即每个数据减去均值, 再除以标准差, 从而使新的数据的均值为0, 方差为1*)输出是{, , , , , , ,, , , , , ,, , , , , , }请验算新的数据的均值是0, 标准差是1.2. 作样本的直方图, 为分布的2χ检验作准备.例2下面列出了84个伊特拉斯坎(Etruscan)人男子的头颅的最大宽度(mm), 对数据分组,并作直方图. 为检验这些数据是否来自正态总体(α=作准备.1 411481321381541421501461551581 501401471481441501491451491581 431411441441261401441421411401 451351471461411361401461421371 481541371391431401311431411491 481351481521431441411431471461 501321421421431531491461491381 421491421371341441461471401421 40137152145解如果本次开机还没有输入调用统计软件包命令, 则首先输入并执行命令<<Statistics`(*因占用内存的原因,在刚刚开机时就应该调用所需软件包*)由于作直方图需要调用作图软件包, 因此提前输入调用作图软件包命令<<Graphics\(*也在刚开机时就输入并执行*) 输入数据data2={141, 148, 132, 138, 154, 142, 150, 146, 155, 158, 150, 140, 147, 148, 144, 150, 149, 145, 149, 158, 143, 141, 144, 144, 126, 140, 144, 142, 141, 140, 145, 135, 147, 146, 141, 136, 140, 146, 142, 137, 148, 154, 137, 139, 143, 140, 131, 143, 141, 149, 148, 135, 148, 152, 143, 144, 141, 143, 147, 146, 150, 132, 142, 142, 143, 153, 149, 146, 149, 138, 142, 149, 142, 137, 134, 144, 146, 147, 140, 142, 140, 137, 152, 145};先求数据的最小和最大值. 输入Min[data2]Max[data2]得到最小值126, 最大值158. 取区间[, ], 它能覆盖所有数据. 将[, ]等分为7个小区间, 设小区间的长度为. 数出落在每个小区间内的数据个数即频数i f , 这可以有BinCount 命令来完成. 输入f1=BinCounts[data2, {, , 5}]输出为{1,4,10,33,24,9,3}输入gc=Table[+j*, {j,1,7}](*产生7个小区间的中心的集合gc*) bc=Transpose[{f1/Length[data2], gc}] (* Length[data2]为数据data2的总个数即样本的容量n , f1/Length[data2]为频率n f i /，Transpose 是求矩阵转置的命令，这样bc 为数据对，第一个数是频率，第二个数是组中心*)输出结果为.}}157281{.}152283{.},147,72{.},142,2811{.},137,425{.},132,211{.},127,841{{，，， 输入作频率n f i /对组中心的条形图命令:BarChart[bc]得到如下的条形图(图:图这个条形图条与条之间有间隙，与习惯不一致. 下面的一段程序是利用Mathematica的作广义条形图命令GeneralizedBarChart所编制的直接作数据的直方图的命令PlotBinData(自定义), 读者只要记住命令PlotBinData的格式是:PlotBinData[数据, 分组时小区间的长度]输入具体的数据(集合的形式)和分组方式后就可以作出所要的直方图了. 先输入小程序:PlotBinData[data_List, incr_ ]:=Module[{min=Floor[Min[data]], max=Ceiling[Max[data]]},Bindata=BinCounts[data, {min, max, incr}]/Length[data];GeneralizedBarChart[Transpose[{Table[min+incr*j-incr/2,{j,1,Length[Bindata]}],Bindata}]/.{a_,b_}->{a, b, incr}, PlotRange->All]];执行后再输入PlotBinData[data2, 5](*对数据data2作直方图, 小区间长度为5*)执行后得到输出图形(图图请保存定义PlotBinData命令的程序, 将来可用于解决作业中的作图问题.四实验作业1．在台湾省的一项《夫妻对电视传播媒介观念差距的研究》中，访问了30对夫妻，其中丈夫所受教育x（单位：年）的数据如下：18,20,16,6,16,17,12,14,16,18,14,14,16,9,20,18,12,15,13,16,16,21,21,9,16,20,14,14,16,16.(1)求样本均值, 中位数, 四分位数; 样本方差, 样本标准差, 极差, 变异系数, 二阶、三阶和四阶中心矩; 求偏度, 峰度.(2)将数据分组，使组中值分别为6，9，12，15，18，21作出x的频数分布表；作出频率分布的直方图；(3)适当选择分组的小区间长度, 用PlotBinData命令作频率直方图.2．下面的数据是有50名大学新生的一个专业在数学素质测验中所得到的分数：88,74,67,49,69,38,86,77,66,75,94,67,78,69,84,50,39,58,79,70,90,79,97,75,98, 77,64,69,82,71,65,68,84,73,58,78,75,89,91,62,72,74,81,79,81,86,78,90,81,62.将这组数据分成6~8个组，画出频率直方图，并求出样本均值、样本方差; 并求偏度, 峰度.。

MathematicsAnswer all questions1.Given that 2x+y= 16 and In (2x–y) = ½ In 5, find the value of x and of y, leaving youranswers in surd form. [8]X=(4+√5)/3 Y=(8-√5)/32.Given that A = {x:-5 < 2x-3 ≤ 15}, B = {x: -4 ≤ x + 1 < 7} andA UB = {x: a ≤ x -1≤ b}, find the value of a and of b. [5]a=-6 b=83.Functions f, g and h are defined byf:x→ 2 + xg:x→ -|x-1|h:x→ -x2 + 2(a)Find the values of gh(-2) and f-1gh(-2).-3and-1 [4](b)Find in similar form, gf, fg and gh. gf=-|x+1| fg=2-|x-1| gh=-|-x2+1|[3](c)Find the values of x such that fg(x) = -1 x=4 or x=-2[3](d)Show that hg(x) = 1 + 2x- x2. ∵hg(x)=h(-|x-1|)=-(-|x-1|)2+2=-(x-1)2+2=-x2+2x+1 [4]4.Three points have coordinates A(5, -3), B(-2,1) and C(a , 5).(a)Find, in terms of a, the coordinates of M, which is the midpoint of BC. [4]The midpoint of BC is ((a-2)/2,3)(b)Find the value of a for which AM is perpendicular to ABa=132/7. [6]5.Find the fifth and sixth terms of binomial expansion of ( 2- ⅓ x2)9. Hence find thecoefficient of x10 in ( 2- ⅓ x2 )9 (3x2+5 ) . 5th:(-19/3)9 6th:( -10) 9[8]6.Find the coordinates of the points of intersection of the line x + y = 3 and the curvex2– 2x +2y2= 3. [6] x=3, y=0 or x=3/5,y=12/57.Find the range of values of k for which the equation x2– 6x + k2– 7 = 0 has real roots. 36-4×1×（k2-7）≥0 the result is -4≤k≤4[6] 8.Find all the angles between 0° and 360° inclusive which satisfysinxsec2x– 2tan x = 0 x=60°[6] 9. A body moves in a straight line so that its displacement, S m, from a point O at time tsec, is given by S = 5 + 15t2– 5t3.Find(a)The time when the body is instantaneously at rest, [4](b)The acceleration when t = 4, Answer:75[3](c)The total distance moved by the body in 9 seconds, [6](d)The average speed of the moving body. [4]10.A semicircle of centre O and radius 10 cm has diameter AB. The chord AQ is10 cm and the ratio of arc AP to arc AQ is 2 : 3.(a)Show that ∠AOP is 2⁄9 π radians.∵AO=QO=AQ=10 cm ∴∠AOQ=60° and∵the ratio of arc AP to arc AQ is2 : 3. ∴∠AOP=40°(40°/360°)×2л=2л/9[6](b)Calculate the area of segment AQP. the area of segment AQP is (50л/3)-25√3 [4]11.If x2– 2x– 3 is a factor of the expression x4+ px3+ qx– 81, find the value of p and ofq. With these of p and q, factorize the expression completely. [10] 数学回答以下所有问题1.已知2x+y和ln（2x-y）= ½ In 5 求x和y的值，保留根号。

Professor & Head of DepartmentNT Bishop, MA(Cambridge), PhD(Southampton), FRASSenior LecturersJ Larena, MSc(Paris), PhD(Paris)D Pollney, PhD(Southampton)CC Remsing, MSc(Timisoara), PhD(Rhodes)Vacant LecturersEOD Andriantiana, PhD(Stellenbosch)V Naicker, MSc(KwaZulu-Natal)AL Pinchuck, MSc(Rhodes), PhD(Wits)Lecturer, Academic Development M Lubczonok, Masters(Jagiellonian)Mathematics (MA T) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as major subjects for the degrees of BSc, BA, BJourn, BCom, BBusSci, BEcon and BSocSc, and for the diploma HDE(SEC).To major in Mathematics, a candidate is required to obtain credit in the following courses: MAT1C; MAM2; MAT3. See Rule S.23.To major in Applied Mathematics, a candidate is required to obtain credit in the following courses: MAT1C, MAM2; MAP3. See Rule S.23.The attention of students who hope to pursue careers in the field of Bioinformatics is drawn to the recommended curriculum that leads to postgraduate study in this area, in which Mathematics is a recommended co-major with Biochemistry, and for which two years of Computer Science and either Mathematics or Mathematical Statistics are prerequisites. Details of this curriculum can be foundin the entry for the Department of Biochemistry, Microbiology and Biotechnology.See the Departmental Web Page http://www.ru.ac.za/departments/mathematics/ for further details, particularly on the content of courses.First-year level courses in MathematicsMathematics 1 (MAT1C) is given as a year-long semesterized two-credit course. Credit in MAT1C must be obtained by students who wish to major in certain subjects (such as Applied Mathematics, MATHEMATICS (PURE AND APPLIED)Physics and Mathematical Statistics) and by students registered for the BBusSci degree.Introductory Mathematics (MAT1S) is recommended for Pharmacy students and for Science students who do not need MAT1C or MAT1C1.Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.Mathematics 1L (MA T1L) is a full year course for students who do not qualify for entry into any of the first courses mentioned above. This is particularly suitable for students in the Social Sciences and Biological Sciences who need to become numerate or achieve a level of mathematical literacy. A successful pass in this course will give admission to MA T1C.First yearMAT1CThere are two first-year courses in Mathematics for candidates planning to major in Mathematics or Applied Mathematics. MAT1C1 is held in thefirst semester and MAT1C2 in the second semester. Credit may be obtained in each course separately and, in addition, an aggregate mark of at least 50%will be deemed to be equivalent to a two-creditcourse MAT1C, provided that a candidate obtains the required sub-minimum (40%) in each component. Supplementary examinations may be recommended in either course, provided that a candidate achieves a minimum standard specified by the department. Candidates obtaining less than 40% for MAT1C1 are not permitted to continue with MAT1C2.MAT1C1 (First semester course): Basic concepts (number systems, functions), calculus (limits,continuity, differentiation, optimisation, curvesketching, introduction to integration), propositional calculus, mathematical induction, permutations, combinations, binomial theorem, vectors, lines andplanes, matrices and systems of linear equations.MAT1C2 (Second semester course): Calculus (integration, applications of integration, improper integrals), complex numbers, differential equations, partial differentiation, sequences and series.MAT1S (Semester course: Introductory Mathematics) (about 65 lectures)Estimation, ratios, scales (log scales), change of units, measurements; Vectors, systems of equations, matrices, in 2-dimensions; Functions: Review of coordinate geometry, absolute values (including graphs); Inequalities; Power functions, trig functions, exponential functions, the number e (including graphs); Inverse functions: roots, logs, ln (including graphs); Graphs and working with graphs; Interpretation of graphs, modeling; Descriptive statistics (mean, standard deviation, variance) with examples including normally distributed data; Introduction to differentiation and basic derivatives; Differentiation techniques (product, quotient and chain rules); Introduction to integration and basic integrals; Modeling, translation of real-world problems into mathematics.MAT 1L: Mathematics Literacy This course helps students develop appropriatemathematical tools necessary to represent and interpret information quantitatively. It also develops skills and meaningful ways of thinking, reasoning and arguing with quantitative ideas in order to solve problems in any given context.Arithmetic: Units of scientific measurement, scales, dimensions; Error and uncertainty in measure values.Fractions and percentages - usages in basic science and commerce; use of calculators and spreadsheets. Algebra: Polynomial, exponential, logarithmic and trigonometric functions and their graphs; modelling with functions; fitting curves to data; setting up and solving equations. Sequences and series, presentation of statistical data.Differential Calculus: Limits and continuity; Rules of differentiation; Applications of Calculus in curvesketching and optimisation.Second Year Mathematics 2 comprises two semesterized courses,MAM201 and MAM202, each comprising of 65 lectures. Credit may be obtained in each course seperately. An aggregate mark of 50% will grant the two-credit course MAM2, provided a sub-minimumof 40% is achieved in both semesters. Each semester consists of a primary and secondary stream which are run concurrently at 3 and 2 lectures per week, respectively. Additionally, a problem-based course in Mathematical Programming contributes to the class record and runs throughout the academic year.MAM201 (First semester):Advanced Calculus (39 lectures): Partial differentiation: directional derivatives and the gradient vector; maxima and minima of surfaces; Lagrange multipliers. Multiple integrals: surface and volume integrals in general coordinate systems. Vector calculus: vector fields, line integrals, fundamental theorem of line integrals, Green’s theorem, curl and divergence, parametric curves and surfaces.Ordinary Differential Equations (20 lectures): First order ordinary differential equations, linear differential equations of second order, Laplace transforms, systems of equations, series solutions, Green’s functions.Mathematical Programming 1 (6 lectures): Introduction to the MATLAB language, basic syntax, tools, programming principles. Applicationstaken from MAM2 modules. Course runs over twosemesters.MAM202 (Second semester):Linear Algebra (39 lectures): Linear spaces, inner products, norms. Vector spaces, spans, linear independence, basis and dimension. Linear transformations, change of basis, eigenvalues, diagonalization and its applications.Groups and Geometry (20 lectures): Number theory and counting. Groups, permutation groups, homomorphisms, symmetry groups in 2 and 3 dimensions. The Euclidean plane, transformations and isometries. Complex numbers, roots of unity and introduction to the geometry of the complex plane.Mathematical Programming 2 (6 lectures): Problem-based continuation of Semester 1.Third-year level courses inMathematics and Applied Mathematics Mathematics and Applied Mathematics are offered at the third year level. Each consists of four modules as listed below. Code TopicSemester Subject AM3.1 Numerical analysis 1 Applied MathematicsAM3.2 Dynamical systems 2 Applied Mathematics AM3.4 Partial differentialequations 1 Applied Mathematics AM3.5 Advanced differentialequations 2 Applied MathematicsM3.1 Algebra 2 MathematicsM3.2 Complex analysis 1 MathematicsM3.3 Real analysis 1 MathematicsM3.4 Differential geometry 2 Mathematics Students who obtain at least 40% in all of the above modules will be granted credit for both MAT3 and MAP3, provided that the average of the Applied Mathematics modules is at least 50% AND the average of the Mathematics modules is at least 50%. Students who obtain at least 40% for any FOUR of the above modules and with an average mark over the four modules of at least 50%, will be granted credit in either MAT3 or MAP3. If three or four of the modules are from Applied Mathematics then the credit will be in MAP3, otherwise it will be in MAT3.Module credits may be carried forward from year to year.Changes to the modules offered may be made from time-to-time depending on the interests of the academic staff.Credit for MAM 2 is required before admission to the third year courses.M3.1 (about 39 lectures) AlgebraAlgebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.Syllabus: Sets, equivalence relations, groups, rings, fields, integral domains, homorphisms, isomorphisms, and their elementary properties.M3.2 (about 39 lectures) Complex Analysis Building on the first year introduction to complex numbers, this course provides a rigorous introduction to the theory of functions of a complex variable. It introduces and examines complex-valued functions of a complex variable, such as notions of elementary functions, their limits, derivatives and integrals. Syllabus: Revision of complex numbers, Cauchy- Riemann equations, analytic and harmonic functions, elementary functions and their properties, branches of logarithmic functions, complex differentiation, integration in the complex plane, Cauchy’s Theorem and integral formula, Taylor and Laurent series, Residue theory and applications. Fourier Integrals.M3.3 (about 39 lectures) Real AnalysisReal Analysis is the field of mathematics that studies properties of real numbers and functions on them. The course places great emphasis on careful reasoning and proof. This course is an essential basis for any further study in mathematics.Syllabus: Topology of the real line, continuity and uniform continuity, Heine-Borel, Bolzano-Weierstrass, uniform convergence, introduction to metric spaces.M3.4 (about 39 lectures) Differential Geometry Roughly speaking, differential geometry is concerned with understanding shapes and their properties in terms of calculus. This elementary course on differential geometry provides a perfect transition to higher mathematics and its applications. It is a subject which allows students to see mathematics for what it is - a unified whole mixing together geometry, calculus, linear algebra, differential equations, complex variables, calculus of variations and topology.Syllabus: Curves (in the plane and in the space), curvature, global properties of curves, surfaces, the first fundamental form, isometries, the second fundamental form, the normal and principal curvatures, the Gaussian and mean curvatures, the Gauss map, geodesics.AM3.1 (about 39 lectures) Numerical Analysis Many mathematical problems cannot be solved exactly and require numerical techniques. These techniques usually consist of an algorithm which performs a numerical calculation iteratively until certain tolerances are met. These algorithms can be expressed as a program which is executed by a computer. The collection of such techniques is called “numerical analysis”.Syllabus: Systems of non-linear equations, polynomial interpolation, cubic splines, numerical linear algebra, numerical computation of eigenvalues, numerical differentiation and integration, numerical solution of ordinary and partial differential equations, finite differences,, approximation theory, discrete Fourier transform.AM3.2 (about 39 lectures) Dynamical Systems This module is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields of applied mathematics (like control theory and the Lagrangian and Hamiltonian formalisms of classical mechanics). The emphasis is on the mathematical aspects of various constructions and structures rather than on the specific physical/mechanical models. Syllabus Linear systems; Linear control systems; Nonlinear systems (local theory); Nonlinear control systems; Nonlinear systems (global theory); Applications : elements of optimal control and/or geometric mechanics.AM3.4 (about 39 lectures) Partial Differential EquationsThis course deals with the basic theory of partial differential equations (elliptic, parabolic and hyperbolic) and dynamical systems. It presents both the qualitative properties of solutions of partial differential equations and methods of solution. Syllabus: First-order partial equations, classification of second-order equations, derivation of the classical equations of mathematical physics (wave equation, Laplace equation, and heat equation), method of characteristics, construction and behaviour of solutions, maximum principles, energy integrals. Fourier and Laplace transforms, introduction to dynamical systems.AM3.5 (about 39 lectures)Advanced differential equationsThis course is an introduction to the study of nonlinearity and chaos. Many natural phenomena can be modeled as nonlinear ordinary differential equations, the majority of which are impossible to solve analytically. Examples of nonlinear behaviour are drawn from across the sciences including physics, biology and engineering.Syllabus:Integrability theory and qualitative techniques for deducing underlying behaviour such as phase plane analysis, linearisations and pertubations. The study of flows, bifurcations, the Poincare-Bendixson theorem, and the Lorenz equations.Mathematics and Applied Mathematics Honours Each of the two courses consists of either eight topics and one project or six topics and two projects.A Mathematics Honours course usually requires the candidate to have majored in Mathematics, whilst Applied Mathematics Honours usually requires the candidate to have majored in Applied Mathematics. The topics are selected from the following general areas covering a wide spectrum of contemporary Mathematics and Applied Mathematics: Algebra; Combinatorics; Complex Analysis; Cosmology; Functional Analysis; General Relativity; Geometric Control Theory; Geometry; Logic and Set Theory; Measure Theory; Number Theory; Numerical Modelling; Topology.Two or three topics from those offered at the third-year level in either Mathematics or Applied Mathematics may also be taken in the case of a student who has not done such topics before. With the approval of the Heads of Department concerned, the course may also contain topics from Education, and from those offered by other departments in the Science Faculty such as Physics, Computer Science, and Statistics. On the other hand, the topics above may also be considered by such Departments as possible components of their postgraduate courses.Master’s and Doctoral degrees in Mathematics or Applied MathematicsSuitably qualified students are encouraged to proceed to these degrees under the direction of the staff of the Department. Requirements for these degrees are given in the General Rules.A Master’s degree in either Mathematics or Applied Mathematics may be taken by thesis only, or by a combination of course work and a thesis. Normally four examination papers and/or essays are required apart from the thesis. The whole course of study must be approved by the Head of Department.。

数学mathematics, maths（BrE）, math（AmE）被除数dividend 除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses（pl.）命题proposition算术arithmetic加plus（prep.）, add（v.）, addition（n.）被加数augend, summand加数addend和sum减minus（prep.）, subtract（v.）, subtraction（n.）被减数minuend减数subtrahend差remainder乘times（prep.）, multiply（v.）, multiplication（n.）被乘数multiplicand, faciend乘数multiplicator积product除divided by（prep.）, divide（v.）, division（n.）整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae（pl.）单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth root n次方根the root of n, the nth root集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition 值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite（a.）infinity（n.）无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number 矩阵matrix行列式determinant 几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi（pl.）, diamond 正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon圆circle圆心centre（BrE）, center（AmE）半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter面积area轨迹locus, loca（pl.）相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron 十二面体dodecahedron 二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incentre（BrE）, incenter（AmE）外心excentre（BrE）, excenter（AmE）旁心escentre（BrE）, escenter（AmE）垂心orthocentre（BrE）, orthocenter（AmE）重心barycentre（BrE）, barycenter（AmE）内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation 比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution 图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagram。

数学mathematics, maths(BrE), math(AmE)公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n次方根倒数（reciprocal）x的倒数为1/x有理数rational number无理数irrational number实数real number虚数imaginary number数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth root n次方根the root of n, the nth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量constant变量variable坐标系coordinates坐标轴x-axis, y-axis, z-axis横坐标x-coordinate纵坐标y-coordinate原点origin象限quadrant截距(有正负之分)intercede（方程的）解solution几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem 钝角三角形obtuse triangle不等边三角形scalene triangle 等腰三角形isosceles triangle等边三角形equilateral triangle 四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width周长perimeter面积area相似similar全等congruent三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant补充：集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition 值域range单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral复数complex number矩阵matrix行列式determinant圆circle圆心centre(BrE), center(AmE) 半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference轨迹locus, loca(pl.)平行六面体parallelepiped立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron二十面体icosahedron多面体polyhedron旋转rotation轴axis球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space双曲线hyperbola抛物线parabola四面体tetrahedron五面体pentahedron六面体hexahedron菱形rhomb, rhombus, rhombi(pl.), diamond 正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon相位phase周期period振幅amplitude内心incentre(BrE), incenter(AmE)外心excentre(BrE), excenter(AmE)旁心escentre(BrE), escenter(AmE)垂心orthocentre(BrE), orthocenter(AmE)重心barycentre(BrE), barycenter(AmE)内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图 pie diagram数学 mathematics, maths(BrE), math(AmE) 公理 axiom定理 theorem 计算 calculation运算 operation 证明 prove假设 hypothesis, hypotheses(pl.) 命题 proposition算术 arithmetic 加 plus(prep.), add(v.), additio n(n.)被加数 augend, summand 加数 addend和 sum 减 minus(prep.), subtract(v.), s ubtraction(n.)被减数 minuend 减数 subtrahend差 remainder 乘 times(prep.), multiply(v.), m ultiplication(n.)被乘数 multiplicand, faciend 乘数 multiplicator积 product 除 divided by(prep.), divide(v.), division(n.)被除数 dividend 除数 divisor商 quotient 等于 equals, is equal to, is equ ivalent to大于 is greater than 小于 is lesser than大于等于 is equal or greater than 小于等于 is equal or lesser than 运算符 operator 数字 digit数 number 自然数 natural number整数 integer 小数 decimal小数点 decimal point 分数 fraction分子 numerator 分母 denominator比 ratio 正 positive负 negative 零 null, zero, nought, nil十进制 decimal system 二进制 binary system十六进制 hexadecimal system 权 weight, significance进位 carry 截尾 truncation四舍五入 round 下舍入 round down上舍入 round up 有效数字 significant digit无效数字 insignificant digit 代数 algebra公式 formula, formulae(pl.) 单项式 monomial多项式 polynomial, multinomial 系数 coefficient未知数 unknown, x-factor, y-factor, z-factor 等式，方程式 equation一次方程 simple equation 二次方程 quadratic equation三次方程 cubic equation 四次方程 quartic equation不等式 inequation 阶乘 factorial对数 logarithm 指数，幂 exponent乘方 power 二次方，平方 square三次方，立方 cube 四次方 the power of four, the fo urth powern次方 the power of n, the nth power 开方 evolution, extraction二次方根，平方根 square root 三次方根，立方根 cube root四次方根 the root of four, the fourth root n次方根 the root of n, the n th root集合 aggregate 元素 element空集 void 子集 subset交集 intersection 并集 union补集 complement 映射 mapping函数 function 定义域 domain, field of definiti on值域 range 常量 constant变量 variable 单调性 monotonicity奇偶性 parity 周期性 periodicity图象 image 数列，级数 series微积分 calculus 微分 differential导数 derivative 极限 limit无穷大 infinite(a.) infinity(n.) 无穷小 infinitesimal积分 integral 定积分 definite integral不定积分 indefinite integral 有理数 rational number无理数 irrational number 实数 real number虚数 imaginary number 复数 complex number矩阵 matrix 行列式 determinant几何 geometry 点 point线 line 面 plane体 solid 线段 segment射线 radial 平行 parallel相交 intersect 角 angle角度 degree 弧度 radian锐角 acute angle 直角 right angle钝角 obtuse angle 平角 straight angle周角 perigon 底 base边 side 高 height三角形 triangle 锐角三角形 acute triangle直角三角形 right triangle 直角边 leg斜边 hypotenuse 勾股定理 Pythagorean theorem钝角三角形 obtuse triangle 不等边三角形 scalene triangle 等腰三角形 isosceles triangle 等边三角形 equilateral triangle 四边形 quadrilateral 平行四边形 parallelogram矩形 rectangle 长 length宽 width 菱形 rhomb, rhombus, rhombi(pl.), diamond正方形 square 梯形 trapezoid直角梯形 right trapezoid 等腰梯形 isosceles trapezoid五边形 pentagon 六边形 hexagon七边形 heptagon 八边形 octagon九边形 enneagon 十边形 decagon十一边形 hendecagon 十二边形 dodecagon多边形 polygon 正多边形 equilateral polygon圆 circle 圆心 centre(BrE), center(AmE)半径 radius 直径 diameter圆周率 pi 弧 arc半圆 semicircle 扇形 sector环 ring 椭圆 ellipse圆周 circumference 周长 perimeter面积 area 轨迹 locus, loca(pl.)相似 similar 全等 congruent四面体 tetrahedron 五面体 pentahedron六面体 hexahedron 平行六面体 parallelepiped立方体 cube 七面体 heptahedron八面体 octahedron 九面体 enneahedron十面体 decahedron 十一面体 hendecahedron十二面体 dodecahedron 二十面体 icosahedron多面体 polyhedron 棱锥 pyramid棱柱 prism 棱台 frustum of a prism旋转 rotation 轴 axis圆锥 cone 圆柱 cylinder圆台 frustum of a cone 球 sphere半球 hemisphere 底面 undersurface表面积 surface area 体积 volume空间 space 坐标系 coordinates坐标轴 x-axis, y-axis, z-axis 横坐标 x-coordinate纵坐标 y-coordinate 原点 origin双曲线 hyperbola 抛物线 parabola三角 trigonometry 正弦 sine余弦 cosine 正切 tangent余切 cotangent 正割 secant余割 cosecant 反正弦 arc sine反余弦 arc cosine 反正切 arc tangent反余切 arc cotangent 反正割 arc secant反余割 arc cosecant 相位 phase周期 period 振幅 amplitude内心 incentre(BrE), incenter(AmE) 外心 excentre(BrE), excenter(AmE) 旁心 escentre(BrE), escenter(AmE) 垂心 orthocentre(BrE), orthocenter (AmE)重心 barycentre(BrE), barycenter(AmE) 内切圆 inscribed circle外切圆 circumcircle 统计 statistics平均数 average 加权平均数 weighted average方差 variance 标准差 root-mean-square deviation, s tandard deviation比例 propotion 百分比 percent百分点 percentage 百分位数 percentile排列 permutation 组合 combination概率，或然率 probability 分布 distribution正态分布 normal distribution 非正态分布 abnormal distribution 图表 graph 条形统计图 bar graph柱形统计图 histogram 折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图 pie diagram。

1.数学mathematics2.算术arithmetic3.代数algebra4.几何geometry5.解析几何Analytic geometry微积分calculus分析analysis概率probability统计statistics拓扑topology自然数Natural number素数primes整数integer有理数Rational number分数fractional number分子numerator (in a fraction) ['nju:məreitə]分母denominator无理数Irrational Numbers实数Real number正数positive number负数negative number非负数nonnegative number复数/虚数complex number / imaginary number 加法additiona +b=c a plusb equals c和sum减法subtractiona-b a minus b差minus乘法timesa*b a multiplied b积product除法divisiona /b a divided by b商quotient倒数reciprocal因数，因子factor零point0.4 0 point four比值specific value（ratio）（十进制）小数位Decimal Places常数constant系数coefficient有限limited\ finite无限infiniteInfinity奇数Odd number uneven number偶数even number等式equality方程equation不等式inequality群Group环ring域domain等价equivalent多项式polynomial线性linear线性相关/无关linear correlation / linear independence 二次方程Quadratic equation矩阵matrix列column行row方（矩）阵square matrix对角线diagonal line行列式determinant秩rank逆inverse单位矩阵identity matrix转置transposition特征值Characteristic value eigenvalue 特征向量Feature vector eigenvector 点point线line面surface空间space平行线parallel lines垂线vertical line切线tangent line斜率rake ratio / slope角angle边side面积area体积volume三角形triangle正方形square矩形rectangular椭圆elliptic抛物线parabolic curve双曲线hyperbolic圆circle圆心the centre of a circle圆周，周长circumference ，perimeter半径radius直径diameter弧arc度degree球ball圆柱体cylinder坐标，坐标系coordinates，Coordinate system 坐标轴coordinate axisx 轴X axis向量vector截距intercept内积/点积Inner product/dot product外积/叉积outer product/ cross product集合set元素element属于belong toa is an element of A空集empty set / null set 子集subset并集union set交集intersection set区间interval开区间Open interval闭区间Closed interval邻域neighborhood映射mapping满射surjection单射Single shot injection 一一映射one-to-one mapping 函数function自变量independent variable 因变量Dependent variable定义域definitional domain值域range复合函数Composite function 逆映射inverse mapping反函数inverse function显函数Explicit function隐函数Implicit function幂power指数exponent指数函数exponential function对数logarithm对数函数Logarithm function三角函数trigonometric function \ Triangle function 有界bounded无界unbounded单调增加Monotone increasing单调减少monotone decreasing周期period最大值maximum value最小值Minimum value极大值maximum value maximal value极小值minimal value凹concave凸convex2x the square of xsquare root of x3x the cube of xcube root of xnx x to the power nnth root of x极限the limit趋向于x approach x approaches the limit infinity 收敛convergence发散divergent唯一uniqueness无穷小indefinitely small无穷大infinity渐近线asymptotic line连续continuous间断discontinuous离散discrete导数derivative（differential cofficient ）微分differential可微differentiable二阶导数Second order derivatives中值定理Mean value theorem驻点Stationary point曲率curvature差分difference不定积分Indefinite integral定积分Definite integral积分号integral number被积函数integrand可积integral换元积分法integration by substitution分部积分法integration by parts微分基本定理Differential basic theorem /fundamental theorem of calculus [文档可能无法思考全面，请浏览后下载，另外祝您生活愉快，工作顺利，万事如意!]。

Mathematica的基本语法特征如果你是第一次使用Mathematica，那么以下几点请你一定牢牢记住：Mathematica中大写小写是有区别的，如Name、name、NAME等是不同的变量名或函数名。

系统所提供的功能大部分以系统函数的形式给出，内部函数一般写全称，而且一定是以大写英文字母开头，如Sin[x],Conjugate[z]等。

乘法即可以用*，又可以用空格表示，如2 3＝2*3＝6 ,x y,2 Sin[x]等；乘幂可以用“^”表示，如x^0.5,Tan[x]^y。

自定义的变量可以取几乎任意的名称，长度不限，但不可以数字开头。

当你赋予变量任何一个值，除非你明显地改变该值或使用Clear[变量名]或“变量名=.”取消该值为止，它将始终保持原值不变。

一定要注意四种括号的用法：()圆括号表示项的结合顺序，如(x+(y^x+1/(2x)));[]方括号表示函数，如Log[x],BesselJ[x,1]；{}大括号表示一个“表”(一组数字、任意表达式、函数等的集合)，如{2x,Sin[12 Pi],{1+A,y*x}}；[[]]双方括号表示“表”或“表达式”的下标，如a[[2,3]]、{1,2,3}[[1]]=1。

Mathematica的语句书写十分方便，一个语句可以分为多行写，同一行可以写多个语句（但要以分号间隔）。

当语句以分号结束时，语句计算后不做输出（输出语句除外），否则将输出计算的结果。

一.数的表示及计算1.在Mathematica中你不必考虑数的精确度，因为除非你指定输出精度，Mathematica总会以绝对精确的形式输出结果。

例如：你输入In[1]:=378/123，系统会输出Out[1]:=126/41，如果想得到近似解，则应输入In[2]:=N[378/123,5],即求其5位有效数字的数值解，系统会输出Out[2]:=3.0732，另外Mathematica还可以根据你前面使用的数字的精度自动地设定精度。

mathematics计算方差方差：定义和计算方差是统计学中衡量数据集离散程度的关键指标。

它表示数据与平均值之间的平均平方差。

计算方差需要以下步骤：步骤 1：计算样本平均值平均值是数据集中所有值的总和除以值的数量。

它用符号μ（mu）表示。

步骤 2：计算每个数据点的偏差偏差是每个数据点与平均值之间的差值。

用符号x - μ表示。

步骤 3：平方每个偏差平方偏差消除偏差的负号，并放大它们的绝对值差异。

平方偏差用符号(x - μ)²表示。

步骤 4：计算平方偏差的总和平方偏差的总和是所有平方偏差的累加。

用符号∑(x - μ)²表示。

步骤 5：除以样本数量-1样本方差是平方偏差总和除以样本数量减去 1。

样本数量用符号n表示。

公式：```样本方差= ∑(x - μ)² / (n - 1)```偏差校正：样本质量的分母为n - 1（而不是n）是为了获得总体方差的无偏估计。

当样本来自总体时，使用n - 1会导致更准确的方差估计。

计算方差的示例：考虑以下数据集：{2, 4, 6, 8, 10}步骤 1：计算平均值μ = (2 + 4 + 6 + 8 + 10) / 5 = 6步骤 2：计算每个偏差x - μ:2 - 6 = -44 - 6 = -26 - 6 = 08 - 6 = 210 - 6 = 4步骤 3：平方每个偏差(x - μ)²:(-4)² = 16(-2)² = 40² = 02² = 44² = 16步骤 4：计算平方偏差的总和∑(x - μ)² = 16 + 4 + 0 + 4 + 16 = 40 步骤 5：除以样本数量-1样本方差 = 40 / (5 - 1) = 10因此，该数据集的样本方差为 10。

mathematics求不定积分的解析式数学是一门研究数量、结构、空间以及变化等概念的学科。

在数学中，求解不定积分是一项非常重要的任务。

不定积分是指求解一个函数的原函数，也被称为反导函数。

求不定积分的解析式是指能够用有限次基本初等函数和常数运算表示的表达式。

在数学中，不定积分的解析式的求解方法有很多种。

下面我将介绍其中的一些常见方法。

最基础的方法是使用基本的积分公式。

例如，我们知道多项式函数的不定积分可以通过逐项求积分得到。

此外，还可以利用三角函数的性质以及指数函数的性质来求解不定积分。

这些基本的积分公式是求解不定积分的基础，熟练掌握它们可以帮助我们更好地解决问题。

当基本的积分公式无法直接求解不定积分时，我们可以尝试使用换元法。

换元法是求不定积分中常用的一种方法，通过引入一个新的变量替代原来的变量，从而将原来的积分变为一个更容易求解的形式。

例如，当我们遇到含有平方根的积分时，可以通过令新的变量等于平方根的形式，将原来的积分转化为一个更简单的形式。

分部积分法也是求解不定积分的常用方法之一。

分部积分法是利用求导和求积分的乘积法则，将原来的积分转化为两个函数的乘积的积分形式。

通过选择合适的两个函数，可以将原来的积分转化为一个更容易求解的形式。

分部积分法在求解一些特殊函数的不定积分时非常有用，例如指数函数和对数函数等。

当我们遇到无法用基本的积分公式、换元法或分部积分法求解的不定积分时，可以尝试使用级数展开法。

级数展开法是将一个函数展开成无限级数的形式，然后对每一项分别进行积分。

通过对级数的求和，可以得到原函数的近似解析式。

级数展开法在一些特殊函数的不定积分求解中非常有用，例如三角函数的不定积分。

还有一些特殊函数的不定积分无法用以上方法求解，需要使用特殊的积分方法。

例如，椭圆积分、贝塞尔函数积分等都需要使用特殊的方法来求解。

这些特殊积分方法是在求解不定积分的过程中逐渐发展起来的，它们在解决特定问题中起到了重要的作用。