综合题新编选登

题 1 2 7　过点 (0 ,1)的直线l与曲线C :y =x+1x (x >0 )交于相异两点 ,设曲线C在这两点处的切线分别为l1与l2 ,求l1与l2 交点的轨迹 .解　设直线l与曲线C交于点M (x1,y1) ,N(x2 ,y2 ) ,l1与l2 交于点P(x ,y) ,直线l的斜率为k ,方程为 y =kx +1.对 y =x +1x求导 ,得 :y′ =1- 1x2 .则 y′|x =x1=1- 1x21,y′|x =x2 =1- 1x21.故直线l1的方程为y - (x1+1x1) =(1- 1x12 ) (x -x1) ,即 y =(1- 1x12 )x +2x1(1)同理 ,可求得l2 的方程为y =(1- 1x22 )x +2x2(2 )(1) - (2 )得 (1x22 - 1x12 )x +2x1- 2x2=0 .由于x1≠x2 ,解得x =2x1x2x1+x2(3)由 …     

直线方程四则运算的意义和价值

1　直线方程的加减运算1 1　意义已知两条直线l1 :A1 x B1 y C1 =0 ,l2 :A2 x B2 y C2 =0 .我们来分析l3:(A1 A2 )x (B1 B2 )y C1 C2 =0和l1 、l2 有什么关系 .( 1 )当l1 ∥l2 时 ,l3也和它们平行 .因为l1 ∥l2 ,有 ,A1 A2 =B1 B2,则 A1 A2A2 =B2 B2B2,所以l3∥l2 .( 2 )当l1 和l2 相交时 .记两直线的交点为P(x0 ,y0 ) ,那么 ,A1 x0 B1 y0 C1 =0和A2 x0 B2 0 C2 =0 ,因此 ,(A1 A2 )x0 (B1 B2 )y0 C1 C2 =0也成立 .所以l3也过点P .我们还可以推广到一般的情况 :直线A1 x B1 y C1 λ(A2 x B2 C2 ) =0…     

非不能也是不为也

题目 :长为 l的线段 P1P2 的两端在抛物线 y =x2上滑动 ,求此线段的中点 M( x,y)的纵坐标的最小值 .解　设线段 P1P2 的中点 M( x,y) ,则由题意易得 :y =14( l21 4 x2 4 x2 ) 1=14[l21 4 x2 ( 1 4 x2 ) - 1 ]2≥ 14[2 l21 4 x2 ( 1 4 x2 ) - 1 ]=l2 - 14(等号当且仅当　 l21 4 x2 =1 4 x2 ,即　 x =± l - 12 时成立 ) .这时 ,只有当 l≥ 1时 ,y才有最小值 l2 - 14.那么 ,当 0 相似文献   

2003年第四期初中数学问题解答

一、分解因式 :6x2 -5xy-4y2 -1 1x 2 2y -1 0 .解 :注意到 6x2 -5xy -4y2 =( 2x y) ( 3x -4y) .设 6x2 -5xy -4y2 -1 1x 2 2y-1 0=( 2x y k) ( 3x -4y l) ,则 6x2 -5xy -4y2 -1 1x 2 2y-1 0=6x2 -5xy -4y2 ( 3k 2l)x ( -4k l)y kl.比较对应项的系数得 :3k 2l=-1 1 ,-4k l=2 2 ,kl=-1 0 .　　解得 k =-5 ,l=2 .于是 6x2 -5xy -4y2 -1 1x 2 2y-1 0　 =( 2x y -5 ) ( 3x -4y 2 ) .二、求函数y =|x2 -4|-3x在区间 -2≤x≤ 5中的最大值和最小值 ,并求当y为最大值时的x值 .解 :若x2 -4≥ 0 ,即 |x|≥ 2 ,则　　y=x2 -3x-4=(x-32 ) 2 -2 54.当 |x|≤ 2时 ,　　y=-x2 -3x 4 =-(x 32 ) 2 2 54.从而求得 :当x=-32 时 ,y最大值 =2 54;当x=...     

周期函数Fourier级数展开式的唯一性

以 2 l为周期的函数 f(x)也可看作周期为 2 kl(k=1 ,2 ,3 ,… ) .设 f(x)满足 Dirichlet充分条件 ,[2 ]证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 4l为周期的 Fourier级数对应的不同表达形式是一致的 .本文则在 [2 ]的基础上 ,进一步证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 2 kl(k=1 ,2 ,3 ,… )为周期的 Fourier级数对应的表达式的一致性 ,从而得出结论 :任一周期函数 f(x)按 [1 ]方法展开的Fourier级数是唯一的 .     

Numerical Analysis of Nonstationary Thermistor Problem

1.AMathematicalModelandaDiscreteSchemeThemodelofanonstationaxythermistorproblemisderivedfromtheconservationlawsofcurrentandenergy(see[l][2]l3]):Findapair{W,u}suchthatV-(a(u)VW)=oinQT=flX(O,T),(1.l)W=Woonoflx(O,T),(1.2)ut-bu=a(U)IVWl'inQT,(1'3)u=oonoflx(O,T),(1.5)u(x,o)=uo(x)infl(1.5)whereflCR"(N21)isaboundeddomain,occupiedbythethermistor;W=yt(x,t),u=u(x,t)aredistributionsoftheelectricalpotentialandthetemperatureinfl,respectively;J(u)isthetemperaturedependentelectricalconductivity;a…     

同时求解f(x)零点的一种迭代解法

1　引　　言在许多实际问题中 ,常常会遇到求解非线性方程 f( x) =0的根 ,或称为求函数 f( x)的零点 .此时 f( x) =( x-α) μg( x) ,且 g(α)≠ 0 ,μ为大于零的常数 ,称为零点α的根指数 .当 f( x)为 n次多项式 ,设 δ(l)k =-f( z(l)k ) /f′( z(l)k ) ,牛顿修正量迭代解法为z(l+1 )k =z(l)k +δ(l)k /( 1 +δ(l)k ni=1 ,i≠ k1z(l)k -z(l)i) ,　　 k =1 ,2 ,… ,n,　 l =0 ,1 ,2 ,… ( 1 )当所有根为单根时 ,迭代法收敛 ,且收敛阶为 3阶 (见 [1 ] ,[2 ] ,[3 ] ,[4 ] ) .当 f ( x)为 n次多项式 ,所有互不相同的根为 r1 ,r2 ,… ,rm,对应…     

BIFURCATION DIAGRAMS OF QUADRATIC DIFFERENTIAL SYSTEMS HMING ONE FOCUS AND ONE SADDLE

In[1]wehavestudiedthebifurcationdiagramsofthequadraticdifferentialsystem-.x=-y 6x lxz mxy y',i=x(1 ax by)inthe(b,m)parameterplanewhena,6andlsatisfythecondition:ao,l 16'(6 l),andwesawsaddleloopcanbifurcateatmosttwolimitcycles(LC,forabbre-viation).Ontheotherhand,inl2]L.A.CherkasandV.A.Gaikohaveprovedthatanyquadraticsystemhavingonlytwofinitecriticalpoints:onefocusandonesaddlepoint,canalwaysbetransformedbynon-degenratelineartransforma-tionsintotheform:whereP1=y(l 7y)-axy-by',Q1=x(…     

连续函数的l凸性

在研究函数的性态时,笔者发现如下定义的l凸函数,它反映了函数中普遍存在的凸偏移现象.定义:设f(x)是定义在实数集D上的实值函数,常数l∈R,若对 xk∈M( D),pk≥ 0,k=1,2,…,n, (n∈N,n≥2),∑nk=1pk=1,都有f(∑ni=1pixi+l)≤∑ni=1pif(xi)则称f(x)为M上的l凸函数;当-f(x)为l凸函数时,称f(x)为M上的l凹函数.下面给出连续函数具有l凸性的两个判定定理:定理 1:设f(x)是定义在 [a,a+2l] (l>0)上的连续的增函数,则f(x)是 [a,a+l]上的l凹函数,也是[a+l,a+2l]上的(-l)凸函数.证明:设xi∈[a,a+l] (i=1,2,…,n),x1≤x2≤…≤xn,则xi+l∈[a+l,a…     

新题征展(30)

A　题组新编1 .( 1 )对任意的 x∈ [- 1 ,1 ],函数 f( x)= x2 - ( k 1 ) x 4的值恒大于 0 ,求实数 k的取值范围 ;( 2 )对任意的 k∈ [- 1 ,1 ],函数 f ( x) =x2 - ( k 1 ) x 4的值恒大于 0 ,求实数 x的取值范围 .2 .( 1 )过点 P( 3,1 )作直线 l交 x、y轴正方向于 A、B点 ,求使△ AOB面积最小时直线l的方程 ;( 2 )过点 P( 3,1 )作直线 l交 x轴正方向于 A,交直线 y =2 x于 B,求使△ AOB面积最小时直线 l的方程 ;( 3)过点 P( 3,1 )作直线 l分别交直线 y= - x - 2与 y =2 x 1于 A、B,且 O′为这两条直线的交点 ,求使△ AO′B面…     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.