C4-1

Partial fractions   1.通分 Transfer two partial fractions into a single fraction   Note:denominators need to be transferred into LCM(lowest common multiple最小公倍数)   2.Transfer a single fraction into two partial fractions Note:分子的degree要比分母小,否则为假分数,可以提出来(C2 lon……

CCC竞赛 2015S3: Gates

  Problem Description For your birthday, you were given an airport. The airport has G gates, numbered from 1 to G. P planes arrive at the airport, one after another. You are to assign the ith plane to permanently dock at any gate 1, . . . , g i (1 ≤ g i ≤ G), at which no previous plane has d……

C3-8

Differential  (又到了最后一节,虽然每本书最后章节都为微积分,但内容难度会逐步加深) 1.Chain rule:(链式法则) Apply to: composite function复合函数 Prove: 2.Product rule:(乘法法则) 注:①本公式可以拓展到三元及以上; ②Quotient rule(除法法则) 可以通过除变乘之后,应用Product rule 3.下面分析几个……

C3-7

Trigonometry 三角函数 1.Addition formulae: ①cos(A-B)=cosAcosB+sinAsinB 证明: Coordinates:P(cosA,sinA);Q(cosB,sinB)     因此cos(A-B)=cosAcosB+sinAsinB。 ②replace B with -B,we get cos(A+B)=cosAcosB-sinAsinB ③replace A with (π/2-A),we get sin(A+B)=sinAcosB+cosAsinB ④replace A with -A,we get sin(A……

CCC竞赛 2015S2: Jerseys

Problem Description A school team is trying to assign jerseys numbered 1, 2, 3, . . . , J to student athletes. The size of each jersey is either small (S), medium (M) or large (L). Each athlete has requested a specific jersey number and a preferred size. The athletes will not be satisfied with a ……

CCC竞赛 2015S1: Zero That Out

Problem Description Your boss has asked you to add up a sequence of positive numbers to determine how much money your company made last year. Unfortunately, your boss reads out numbers incorrectly from time to time. Fortunately, your boss realizes when an incorrect number is read and says “zero”, ……

C3-6

Trigonometry 三角函数 1、New function: ①正割: 图像:(中间为正弦曲线,上下为反函数图像)本图为画反函数过程 ②余割: 图像:本图为完整的反函数 ③余切: 图像: 注:画图原则: 拓展:由定义可得: 2、the inverse function of  y=sinx 图像: 注:即对调x、y轴。 拓展: 经典一题 1、f(x)为二次函数……

C3-4

Numerical methods 数值方法(极限思想) (本章内容不多,主要探究如何锁定并缩小解的范围,属于极限思想) 1、零点定理:如果f(a)*f(b)<0,则在(a,b)中有解(不一定是一个)。 证明:函数从f(a)到f(b)必要穿过x轴,即零点。 2、iteration迭代: 将f(x)=0,转化为x=g(x),之后将一个初始x值 代入iter……

C3-3

The exponential function&log function 指数对数函数 1、Exponential function:     注:①底数a>0,当a=1时,为过(0,1)的常数函数。 ②所有指数函数有且仅有(0,1)交点。 ③若y=a^x,g=b^x,两图像对称,则a与b互为倒数。 ④若a>1,则a越大,图像越靠近y轴;若0<a<1,则a越小越靠近y轴。(可利用③来考虑……