Mathematics
mathematics的意思用法总结
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
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）
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课件
随着时间的变化，Mathematica在许多重要领域得到了 广泛的应用，如金融、生物学等。
现在，它已经被应用于科学的各个领域。
1.0 Mathematica的历史
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许多世界顶尖科学家都是它的忠实支持者。它在许多重 要的发现中扮演着关键的角色，并是数以千计的科技文章 的基石。
在工程中，Mathematica已经成为开发和制造的标准。 世界上许多重要的新产品在它们的设计某一阶段或其它阶 段都依靠了Mathematica的帮助。
在商业上，Mathematica在复杂的金融模型中扮演了重 要的角色，广泛地应用于规划和分析。
同时，Mathematica也被广泛应用于计算机科学和软件 发展：它的语言元件被广泛地用于研究原型和界面环境。
1.0 Mathematica的历史
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当 Mathematica1.0 版 发 布 时 ， 《 纽 约 时 代 报 》 写 道 ： “这个软件的重要性不可忽视”；《商业周刊》将 Mathematica 评 比 为 当 年 十 大 最 重 要 产 品 。 在 科 技 界 ， Mathematica被形容为智能和实践的革命。
1.1 Mathenmtica系统简单操作
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1.1.2 工作窗口操作
工作窗口是用户输入、输出、显示各种信息，以及运行 各种程序的场地，用户的全部操作都将在这里进行，人们 将这种类型的窗口称之为Notebook。
下面举例说明在工作窗口中怎样进行操作。 【例1-1】已知a=2，b=3.7，试求c=a+b的值。 在Untitled-1工作窗口中直接键入：
Mathematica 1.2 (1989) Mathematica 2.1 (1992) Mathematica 3.0 (1996) Mathematica 4.1 (2000) Mathematica 5.0 (2003) Mathematica 5.2 (2005) Mathematica 6.0.3 (2008)
mathematics的意思用法总结
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
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。
Mathematics统计数据
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.。
Mathematics数学-附答案(1)
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。
数学的发展史
数学对人类的重要性
）
就，出现了许多闻名世界的数学家，如刘徽、祖冲之、 王孝通、李冶、秦九韶、朱世杰等人。出现了许多专 门的数学著作，特别是《九章算术》的完成，标志着 我国的初等数学已形成了体系。这部书不但在中国数 学史上而且在世界数学史上都占有重要的地位，一直 受到中外数学史家的重视。我国传统数学在线性方程 组、同余式理论、有理数开方、开立方、高次方程数 值解法、高阶等差级数以及圆周率计算等方面，都长 期居世界领先地位。
这个时期的起点是笛卡尔的著作，他引
这个时期是科学技术
飞速发展的时期，不 断出现震撼世界的重 大创造与发明。二十 世纪的历史表明，数 学已经发生了空前巨 大的飞跃，其规模之 宏伟，影响之深远， 都远非前几个世纪可 比，目前发展处于不 断加速的趋势。
从历史上看，远在巴比伦、埃及时代，由于人类生活和劳动生产的需要积累了一系列 算术和几何的知识。经过希腊时代，将这些比较零散的知识上升为理论的系统。西方
3 、变量数学 入了变量的概念。这个时期中还创立了 一系列新领域：解析几何、微积分、概 时期（十七世 率论、射影几何和数论等。并且出现了 代数化的趋势。随着数学新分支的创立， 新的概念层出不穷，如无理数、虚数、 纪初到十九世 导数、积分等等。 十八世纪是数学蓬勃发展的时期。以微 纪末） 积分为基础发展出一门宽广的数学领
MATHEMATICS（PUREANDAPPLIED）：（纯粹数学和应用）
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.。
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Mathematics
Mathematics (from Greek μ?θημαmáthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.
Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." French mathematician Claire V oisin states "There is creative drive in mathematics, it's all about movement trying to express itself."
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.。