Equation Grapher简体中文版教程(深圳数学教师李红权编写)

2.2直接证明与间接证明2.2.1综合法和分析法学习目标：1.理解综合法、分析法的意义，掌握综合法、分析法的思维特点．(重点、易混点)2.会用综合法、分析法解决问题．(重点、难点)[自主预习·探新知]1．综合法思考1：综合法与分析法的推理过程是合情推理还是演绎推理？[提示]综合法与分析法的推理过程是演绎推理，因为综合法与分析法的每一步推理都是严密的逻辑推理，从而得到的每一个结论都是正确的，不同于合情推理中的“猜想”．思考2: 综合法与分析法有什么区别？[提示]综合法是从已知条件出发，逐步寻找的是必要条件，即由因导果；分析法是从待求结论出发，逐步寻找的是充分条件，即执果索因．[基础自测]1．思考辨析(1)综合法是执果索因的逆推证法．()(2)分析法就是从结论推向已知．()(3)所有证明的题目均可使用分析法证明．()[答案](1)×(2)×(3)×2．命题“对于任意角θ，cos4θ－sin4θ＝cos 2θ”的证明：“cos4θ－sin4θ＝(cos2θ－sin2θ)(cos2θ＋sin2θ)＝cos2θ－sin2θ＝cos 2θ”，其过程应用了()【导学号：48662070】A．分析法B．综合法C．综合法、分析法综合使用D．间接证法B[从证明过程来看，是从已知条件入手，经过推导得出结论，符合综合法的证明思路．]3．要证明A>B，若用作差比较法，只要证明________．A－B>0[要证A>B，只要证A－B>0.]4．将下面用分析法证明a2＋b22≥ab的步骤补充完整：要证a2＋b22≥ab，只需证a2＋b2≥2ab，也就是证________，即证________，由于________显然成立，因此原不等式成立．a2＋b2－2ab≥0(a－b)2≥0(a－b)2≥0[用分析法证明a2＋b22≥ab的步骤为：要证a2＋b22≥ab成立，只需证a2＋b2≥2ab，也就是证a2＋b2－2ab≥0，即证(a－b)2≥0.由于(a－b)2≥0显然成立，所以原不等式成立．][合作探究·攻重难](1)已知a，b是正数，且a＋b＝1，证明：1a＋1b≥4.(2)在△ABC中，a，b，c分别为内角A，B，C的对边，且2a sin A＝(2b－c)sin B ＋(2c－b)sin C.①求证：A的大小为π3；②若sin B＋sin C＝3，证明△ABC为等边三角形.【导学号：48662071】[证明](1)法一：因为a，b是正数且a＋b＝1，所以a＋b≥2ab，所以ab≤12，所以1a＋1b＝a＋bab＝1ab≥4.法二：因为a，b是正数，所以a＋b≥2ab>0，1 a ＋1b≥21ab>0，所以(a＋b)⎝⎛⎭⎪⎫1a＋1b≥4.又a＋b＝1，所以1a ＋1b≥4.法三：1a＋1b＝a＋ba＋a＋bb＝1＋ba＋ab＋1≥2＋2ba·ab＝4.当且仅当a＝b时，取“＝”号．(2)①由2a sin A＝(2b－c)sin B＋(2c－b)sin C，得2a2＝(2b－c)b＋(2c－b)c，即bc＝b2＋c2－a2，所以cos A＝b2＋c2－a22bc＝12，所以A＝π3.②因为A＋B＋C＝180°，所以B＋C＝180°－60°＝120°.由sin B＋sin C＝3，得sin B＋sin( 120°－B)＝3，sin B＋(sin 120°cos B－cos 120°sin B)＝3，32sin B＋32cos B＝3，即sin (B＋30°)＝1.因为0°<B<120°.所以30°<B＋30°<150°，所以B＋30°＝90°，B＝60°.所以A＝B＝C＝60°，即△ABC为等边三角形．1．如图2-2-1所示，在四棱锥P-ABCD中，P A⊥底面ABCD，AB⊥AD，AC⊥CD，∠ABC＝60°，P A＝AB＝BC，E是PC的中点．图2-2-1(1)证明：CD⊥AE；(2)证明：PD⊥平面ABE.[证明](1)在四棱锥P-ABCD中，∵P A⊥底面ABCD，CD⊂平面ABCD，∴P A⊥CD.∵AC⊥CD，P A∩AC＝A，∴CD⊥平面P AC.而AE ⊂平面P AC ，∴CD ⊥AE .(2)由P A ＝AB ＝BC ，∠ABC ＝60°，可得AC ＝P A . ∵E 是PC 的中点，∴AE ⊥PC . 由(1)知，AE ⊥CD ，又PC ∩CD ＝C ， ∴AE ⊥平面PCD .而PD ⊂平面PCD ，∴AE ⊥PD . ∵P A ⊥底面ABCD ，∴PD 在底面ABCD 内的射影是AD . 又AB ⊥AD ，∴AB ⊥PD .又∵AB ∩AE ＝A ，∴PD ⊥平面ABE .【导学号：48662072】[证明] 当a ＋b ≤0时，∵a 2＋b 2≥0，∴a 2＋b 2≥22(a ＋b )成立．当a ＋b ＞0时， 用分析法证明如下：要证a 2＋b 2≥22(a ＋b )，只需证(a 2＋b 2)2≥⎣⎢⎡⎦⎥⎤22(a ＋b )2.即证a 2＋b 2≥12(a 2＋b 2＋2ab )，即证a 2＋b 2≥2ab . ∵a 2＋b 2≥2ab 对一切实数恒成立，∴a2＋b2≥22(a＋b)成立．综上所述，不等式得证．2．已知a，b是正实数，求证：ab＋ba≥a＋b.【导学号：48662073】[证明]要证ab＋ba≥a＋b，只要证a a＋b b≥ab·(a＋b)．即证(a＋b－ab)(a＋b)≥ab(a＋b)，因为a，b是正实数，即证a＋b－ab≥ab，也就是要证a＋b≥2ab，即(a－b)2≥0.而该式显然成立，所以ab ＋ba≥a＋b.[探究问题]1．在实际解题时，综合法与分析法是否可以结合起来使用？提示：在实际解题时，常常把分析法和综合法结合起来使用，即先利用分析法寻找解题思路，再利用综合法有条理地表述解答过程．2．你会用框图表示综合法与分析法交叉使用时的解题思路吗？ 提示：用框图表示如下：其中P 表示已知条件、定义、定理、公理等，Q 表示要证明的结论．已知a 、b 、c 是不全相等的正数，且0<x <1. 求证：log x a ＋b 2＋log x b ＋c 2＋log x a ＋c2<log x a ＋log x b ＋log x c .思路探究：解答本题的关键是利用对数运算法则和对数函数性质转化成整式不等式证明．[证明] 要证明：log x a ＋b 2＋log x b ＋c 2＋log x a ＋c2<log x a ＋log x b ＋log x c ， 只需要证明log x ⎝ ⎛⎭⎪⎫a ＋b 2·b ＋c 2·a ＋c 2<log x (abc )．由已知0<x <1，只需证明a ＋b 2·b ＋c 2·a ＋c2>abc . 由公式a ＋b 2≥ab >0，b ＋c 2≥bc >0，a ＋c2≥ac >0， 又∵a ，b ，c 是不全相等的正数，∴a＋b2·b＋c2·a＋c2>a2b2c2＝abc.即a＋b2·b＋c2·a＋c2>abc成立．∴log x a＋b2＋log xb＋c2＋log xa＋c2<log x a＋log x b＋log x c成立．1．欲证2－3<6－7成立，只需证()A．(2－3)2<(6－7)2B．(2－6)2<(3－7)2C．(2＋7)2<(3＋6)2D．(2－3－6)2<(－7)2C[∵2－3<0，6－7<0，故2－3<6－7⇔2＋7<3＋6⇔(2＋7)2<(3＋6)2.] 2. 在△ABC中，若sin A sin B<cos A cos B，则△ABC一定是()【导学号：48662074】A ．直角三角形B ．锐角三角形C ．钝角三角形D ．等边三角形C [由sin A sin B <cos A cos B 得cos(A ＋B )＝－cos C >0，所以cos C <0，即△ABC 一定是钝角三角形．]3．如果a a ＋b b ＞a b ＋b a ，则实数a ，b 应满足的条件是________． a ≠b 且a ≥0，b ≥0 [a a ＋b b ＞a b ＋b a ⇔a a －a b ＞b a －b b ⇔a (a －b )＞b (a －b ) ⇔(a －b )(a －b )＞0⇔(a ＋b )(a －b )2＞0， 只需a ≠b 且a ，b 都不小于零即可．]4．设a ＞0，b ＞0，c ＞0，若a ＋b ＋c ＝1，则1a ＋1b ＋1c 的最小值为________． 9 [因为a ＋b ＋c ＝1，且a ＞0，b ＞0，c ＞0，所以1a ＋1b ＋1c ＝a ＋b ＋c a ＋a ＋b ＋c b ＋a ＋b ＋c c ＝3＋b a ＋a b ＋c b ＋b c ＋a c ＋c a ≥3＋2b a ·ab ＋2c b ·bc ＋2c a ·ac ＝3＋6＝9.当且仅当a ＝b ＝c 时等号成立．]5．设a ≥b >0，求证：3a 3＋2b 3≥3a 2b ＋2ab 2.(请用分析法和综合法两种方法证明)【导学号：48662075】[证明] 法一：3a 3＋2b 3－(3a 2b ＋2ab 2)＝3a 2(a －b )＋2b 2(b －a )＝(3a 2－2b 2)(a －b )．因为a ≥b >0，所以a －b ≥0,3a 2－2b 2>0，从而(3a 2－2b 2)(a －b )≥0， 所以3a 3＋2b 3≥3a 2b ＋2ab 2.法二：要证3a 3＋2b 3≥3a 2b ＋2ab 2，只需证3a 2(a －b )－2b 2(a －b )≥0，只需证(3a2－2b2)(a－b)≥0，∵a≥b>0.∴a－b≥0,3a2－2b2>2a2－2b2≥0，∴上式成立．。

2012年“感动‘二实’年度人物”候选人事迹材料1.校毽球队关键词：体坛明星，享誉世界清晨，人们还在睡梦中，他们早已开始了一小时的晨练；傍晚，无论严寒酷暑，他们始终坚持训练三小时；周六、周日，他们从不休息，进行从难、从严、从实战出发的适度极限训练；为了中华毽球的传承、弘扬和腾飞，他们甘吃苦中苦，愿求苦中“乐”；他们中先后有8人次获得世界冠军，140多人次获全国冠军……2012年他们发挥敢打敢拼的运动精神，以破竹之势在国内外的毽球赛场上，赛出了风格和水平：广东省第十届中学生运动会毽球赛包揽了全部14枚金牌，广东省毽球锦标赛获得5金6银6铜，全国毽球锦标赛获4金5银3铜，在法国举行的国际毽球公开赛喜获冠亚军。

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2.有效教学团队（高二、初二有效教学教师）关键词：改革先锋，教坛亮点有效教学团队是我校有效教学课堂模式改革的先锋队，它的成立凝聚了校长、专家、部门领导和高二、初二年级教师的心血。

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有效教学的“高效课堂”已逐渐成为学校的一个亮点，学生已成为课堂学习的主体，并且积累了一定“有用”的资料，开发了大量的学习工具单，有效教学的行动还激发了老师们积极探索教改的热情。

NASA Technical Memorandum4435 Hypersonic Lateral and Directional Stability Characteristics of Aeroassist Flight Experiment Configuration in Air and CF4John R.Micol and William L.WellsMAY1993NASA Technical Memorandum4435 Hypersonic Lateral and Directional Stability Characteristics of Aeroassist Flight Experiment Configuration in Air and CF4John R.Micol and William L.WellsLangley Research CenterHampton,VirginiaSummaryThe proposed Aeroassist Flight Experiment (AFE)utilized a14-ft-diameter raked and blunted elliptical cone to demonstrate the ight character-istics of space transfer vehicles(STV's).The AFE was to be carried to orbit by and launched from the Space Shuttle orbiter,where instrumentation for 10on-board experiments would have obtained aero-dynamic and aerothermodynamic data for velocities near32000ft/sec at altitudes above245000ft.A pre ight ground-based test program was initiated to assess the aerodynamic and aerothermodynamic characteristics of the baseline concept and to pro-vide benchmark data for calibration of computational uid dynamics codes to be used in ight predictions. The data reported herein are results from one phase of this ground-based study.Static lateral and di-rectional stability characteristics were obtained for the AFE con guration at angles of attack from010 to10 .Tests were conducted in air at Mach num-bers of6and10and in tetra uoromethane(CF4) at Mach6to examine the e ects of Mach number, Reynolds number,and normal-shock density ratio.Changes in Mach number from6to10in air or in Reynolds number by a factor of4at Mach6had a negligible e ect on the lateral and directional sta-bility characteristics of the baseline AFE con gura-tion.Variations in density ratio across the normal portion of the bow shock from approximately5(air) to12(CF4)had a measurable e ect on lateral and di-rectional aerodynamic coe cients,but no signi cant e ect on lateral and directional stability character-istics.The tests in air and CF4indicated that the con guration was laterally and directionally stable through the test range of angle of attack.Unfortunately,the AFE program was cancelled in late1991.The realization of an AFE ight in the future is possible but uncertain.Thus,this paper documents the lateral and directional aerodynamic characteristics of the baseline AFE vehicle for use in the design of future aeroassist space transfer vehicles. IntroductionAmong the space transportation systems pro-posed for the future are space transfer vehicles (STV's),which are designed to ferry cargo between higher Earth orbits(for example,geosynchronous and lunar orbits)and lower Earth orbit where the Space Shuttle and Space Station Freedom will op-erate.(This class of vehicle was formerly referred to as orbital transfer vehicles or OTV's.)Upon re-turn of the vehicle from high Earth orbit,its velocity must be greatly reduced to attain a nearly circular low Earth orbit.This decrease in velocity can be achieved either by using retrorockets or by guiding the vehicle through a portion of the atmosphere and allowing aerodynamic drag forces to slow the vehi-cle.Studies have shown that lower propellant loads would be required for the aeroassist method(ref.1); thus,payloads could be increased.Future STV's that will be designed to use Earth atmosphere for deceleration are generally referred to as aeroassisted space transfer vehicles or ASTV's (formerly AOTV's).These vehicles will have high drag and a relatively low lift-to-drag ratio and will y at very high altitudes and velocities throughout the atmospheric portion of the trajectory.Before the actual ight vehicle can be designed with optimal aerodynamic and aerothermodynamic characteris-tics,additional information about very high-altitude, high-velocity ight is required.To obtain such in-formation,a subscale ight was proposed whereby a14-ft-diameter ASTV con guration with10on-board experiments would be launched from the Space Shuttle and accelerated back into the atmosphere with a rocket.This Aeroassist Flight Experiment (AFE)would make a sweep through the atmosphere to an altitude of about245000ft with a velocity of nearly32000ft/sec to gain aerodynamic and aero-thermal information and return to low Earth orbit for retrieval by the Space Shuttle.The on-board in-strumentation would measure and record the aero-dynamic characteristics and aerothermodynamic en-vironment of this entry trajectory,and the data would be used to validate computational uid dy-namics(CFD)computer codes and ground-to- ight extrapolation of experimental data for use in future ASTV designs.This ight experiment was proposed because the high-velocity,low-density ow environ-ment cannot be duplicated or simulated in present test facilities,nor can it be predicted with certainty by existing techniques.Naturally,the AFE would require an extensive aerodynamic and aerothermodynamic experimental and computational data base for its design and suc-cessful ight.Present test facilities,in conjunction with the best CFD codes,would provide this infor-mation.For this reason,a pre ight test program in ground-based hypersonic facilities(ref.2)was initiated to develop the required aerodynamic and aerothermodynamic data base.This data base will be used to perform the rst phase of CFD computer code calibration.The experimental results presented herein are part of an extensive ground-based test program performed at the Langley Research Center. Previous results are presented in references3{6.The details of the rationale for the ight experiment areoutlined in reference7,and the set of experiments to be performed is described in reference8.A primary concern for the AFE vehicle is the aerothermal heating on the fore-and aftbody thermal protection system(TPS).Because of these aerother-mal concerns,low values of sideslip angles are desir-able to minimize heating to the aftbody or payload and to prevent large thermal uctuations on the heat shield.Thus,an accurate knowledge of the lateral and directional stability characteristics of the AFE is required.(Lateral and directional stability require-ments for a low lift-to-drag aeromaneuvering vehicle are discussed in ref.9.)CFD codes are not generally used to provide aero-dynamic information for vehicles at sideslip angles. Computed lateral and directional stability charac-teristics for the AFE would require calculations of the entire body at various sideslip angles,thus in-creasing computational time,complexity,and cost. Hence,determination of these stability characteris-tics for the ight vehicle must rely on experimental data obtained in ground-based facilities.This paper addresses the e ects of Mach number, Reynolds number,and normal-shock density ratio(a \real gas"simulation parameter)on lateral and direc-tional aerodynamic characteristics measured on the baseline AFE con guration.Tests were conducted at Mach6and10in air and at Mach6in tetra- uoromethane(CF4)through a range of angle of at-tack and sideslip.During the continuum- ow portion of the ight, the AFE vehicle is expected to undergo normal-shock density ratios of about18,whereas conventional hy-personic wind tunnels that use air or nitrogen as the test gas only produce ratios of5to7.In ight,this large density ratio results from dissociation of air as it passes into the high-temperature shock layer.This real-gas e ect may have a signi cant impact on shock detachment distance,distributions of heating and pressure,and aerodynamic characteristics(ref.10).For blunt bodies at hypersonic speeds,the pri-mary factor that governs the shock stand-o distance and inviscid forebody ow is the normal-shock den-sity ratio.(See ref.10.)Certain aspects of a real gas can be simulated by the selection of a test gas that has a low ratio of speci c heats and provides large values of density ratio.These conditions can be obtained in the Langley Hypersonic CF4Tun-nel,which provides a simulation of this phenomenon by producing a density ratio of about12across the shock.This tunnel,in conjunction with the Lang-ley20-Inch Mach6Tunnel,provides the capability to test a given model at the same free-stream Mach number and Reynolds number,but at two values of density ratio(5.25in air and12.0in CF4).Thus, data for code calibration are provided that include the e ects of normal-shock density ratio.Tests were performed in air at Mach10and through a range of Reynolds numbers at Mach6to verify that aerody-namic characteristics were independent of signi cant changes in Mach numbers and Reynolds numbers for the blunt AFE con guration in hypersonic contin-uum ow.However,the AFE program cancellation ended the research e orts on this con guration.Thus, this paper documents the lateral and directional characteristics of the baseline AFE vehicle for use in the design of future aeroassist space transfer vehicles. SymbolsC l rolling-moment coe cient,Rolling momentq1dSC l=1C l=1 ;per degC n yawing-moment coe cient,Yawing momentq1dSC n =1C n=1 ,per degC y side-force coe cient,Side forceq1SC y=1C y=1 ,per degd model length in symmetry plane,in.M Mach numberp pressure,psiaq dynamic pressure,psiaRe1unit free-stream Reynoldsnumber,ft01Re2;d postshock Reynolds numberbased on dS reference area,model base area,in2(10.604in2when d=3.67in.and4.936in2when d=2.50in.)T temperature, RU velocity,ft/secX moment transfer distance in axialdirection( g.4),in.(1.673in.when d=3.67in.and1.559in.when d=2.50in.)x;y;z axial,lateral,and vertical coordi-nates for AFE( g.4)2Z moment transfer distance innormal direction( g.4),in.(0.129in.when d=3.67in.and0.0979in.when d=2.50in.)angle of attack,degangle of sideslip,degratio of speci c heats of the testgasdensity of the test gas,lbm/in3 Subscripts:t total conditions1free-stream conditions2conditions behind the normalshockAFE Con gurationThe AFE ight vehicle would consist of a14-ft-diameter drag brake,an instrument carrier at the base,a solid-rocket propulsion motor,and small control motors.A sketch of the vehicle is shown in gure1.The drag brake( g.2),which is the forebody con guration,is derived from a blunted 60 half-angle elliptical cone that is raked at73 to the cone centerline to produce a circular raked plane.A skirt with an arc radius equal to one-tenth the rake-plane diameter and with an arc length corresponding to60 has been attached to the rake plane to reduce aerodynamic heating around the base periphery.The blunt nose is an ellipsoid with an ellipticity equal to2.0in the symmetry plane.The ellipsoid nose and the skirt are at a tangent at their respective intersections to the elliptical cone surface.A detailed description of the forebody analytical shape is presented in reference11.Apparatus and TestsFacilitiesLangley31-Inch Mach10Tunnel.The Langley31-Inch Mach10Tunnel(formerly the Lang-ley Continuous Flow Hypersonic Tunnel)expands dry air through a three-dimensional contoured nozzle to a31-in-square test section to achieve a nominal Mach number of10.The air is heated to approxi-mately1850 R by an electrical resistance heater,and the maximum reservoir pressure is approximately 1500psia.The tunnel operates in the blowdown mode with run times of approximately60sec.Force and moment data can be obtained through a range of angle of attack or sideslip during one run by uti-lization of the pitch-pause capability of the model support system.This tunnel is described in more detail in reference12.Langley20-Inch Mach6Tunnel.The20-Inch Mach6Tunnel is a blowdown wind tunnel that uses dry air as the test gas.The air may be heated to a maximum temperature of approximately1100 R by an electrical resistance heater;the maximum reser-voir pressure is525psia.A xed-geometry,two-dimensional,contoured nozzle with parallel side walls expands the ow to a Mach number of6at the20-in-square test section.The model injection mechanism allows changes in angle of attack and sideslip during a run.Run durations are usually60to120sec,al-though longer times can be attained by connection to auxiliary vacuum storage.A description of this facility and the calibration results are presented in reference13.Langley20-Inch Mach6CF4Tunnel.The 20-Inch Mach6CF4Tunnel is a blowdown wind tunnel that uses CF4as the test gas.The CF4 can be heated to a maximum temperature of1530 R by two molten lead bath heat exchangers connected in parallel.The maximum pressure in the tunnel reservoir is2600psia.Flow is expanded through an axisymmetric,contoured nozzle designed to generate a Mach number of6at the20-in-diameter exit.This facility has an open-jet test section.Run duration can be as long as30sec,but10sec is su cient for most tests because the model injection system is not presently capable of changing angle of attack or sideslip during a run.A detailed description of the20-Inch Mach6CF4tunnel is presented in reference14.Just before the present test series,the tunnel was modi ed extensively.Included in those modi cations were a new nozzle,a new test section and model in-jection system,a new di user,and improvements in wiring of the controls and of the data acquisition system.The new nozzle was designed to improve ow quality along the centerline and to more closely match the Mach number in the Mach6air tunnel that is often used to produce data for comparison with the CF4data.Calibration results(ref.15)that were obtained after the new nozzle was installed indi-cate greatly improved ow uniformity near the nozzle centerline.For the present test series,the model was tested on the tunnel centerline.Previously,models were tested o centerline to avoid ow disturbances. (See ref.14.)3ModelsTwo aerodynamic models were fabricated and tested.The models were identical except for size;the base heights(d in g.2)at the symmetry plane were 3.67in.(2.2percent scale)as shown in gure3(a)and 2.50in.(1.5percent scale)as shown in gure3(b). The3.67-in-diameter model is made in three parts| a stainless steel forebody(aerobrake),an aluminum aftbody(instrument carrier and propulsion motor), and a stainless steel balance holder.The2.50-in-diameter model,shown mounted in the Langley 20-Inch Mach6CF4Tunnel in gure3(c),is fabri-cated of aluminum and does not include the circu-lar or hexagonally shaped aftbody and the simulated propulsion motor of previous models that were tested (ref.16).A cylinder protrudes from the base to ac-cept the balance.The acute angle between the bal-ance and cylinder axis and the base in the symmetry plane is73 .The2.50-in-diameter model was fabri-cated to provide an air gap between the end of the balance and the end of the cavity in the forebody; its purpose was to reduce conductive heating.For both models,shrouds were built to shield the bal-ance from base- ow closure.The shrouds attach to the sting,and clearance was provided to avoid in-terference with the balance during model movement when forces and moments were applied.The fore-bodies were machined to the design size and shape within a tolerance of60.003in.Angle of attack(see g.2)and sideslip(see g.4)in this paper are refer-enced to the axis of the original elliptical cone.InstrumentationAerodynamic force and moment data were mea-sured with sting-supported,six-component,water-cooled,internal strain gauge balances.Two ther-mocouples were installed in the water jacket that surrounds the measuring elements to monitor inter-nal balance temperatures.The load rating for each component of the two balances(one for each model size)is presented in table I.The calibration accuracy is0.5percent of the maximum load rating for each component.Test ConditionsThe tests were conducted at nominal free-stream Mach numbers of6and10in air and at Mach6 in CF4.(Nominal test conditions are presented in table II.)The angles of attack for Mach6in air were 0 and65 with nominal sideslip angles of0 ,02 , and04 .Tests at Mach6in CF4were at angles of attack of0 ,65 ,and610 with nominal sideslip angles of0 ,62.5 ,and65 ;at Mach10(except for =02:5 ,where only a negative sweep was performed),the angles of attack were0 ,62.5 ,65 , and610 with nominal sideslip angles of0 ,62 ,and 64 .Test ProceduresBlunt models are conducive to heat conduction through the forebody face during a run,which gener-ally produces a gradual increase in temperature gra-dients along the balance even though the balance is water cooled.Because temperature gradients were not accounted for in the laboratory calibration of the balance,e orts were made to minimize these gradi-ents by limiting the test times.In the20-Inch Mach6 CF4Tunnel,the model was mounted at the desired angle of attack and sideslip before the run.After the test-stream ow was established,the model was in-jected to the test-stream centerline.Data were gath-ered for approximately5sec,then the model was re-tracted.In the air tunnels,the model was mounted at = =0 before the run.After test-stream ow was established,the model was injected to the stream centerline,then pitched to the next angle of attack(or sideslip angle)by the pitch-pause mech-anism.Data were taken while the model was sta-tionary at each position.The balance thermocouples were monitored during each run to assure that the temperature gradient within the balance remained within an acceptable limit.Typical run times for a set of and sweeps in the air facilities were about 15sec.Data Reduction and UncertaintyEach of the three test facilities has a dedicated stand-alone data system.Output signals from the balances were sampled and digitized by an analog-to-digital converter,then stored and processed by a computer.The analog signals were sampled at a rate of50per second in the Mach6CF4and Mach10air tunnels and at20per second in the Mach6air tunnel.A single value of data reported herein represents an average of values measured for 2sec in the Mach6CF4and Mach6air tunnels and for0.5sec in the Mach10air tunnel.Corrections were made for model tare weights at each angle of attack and for interactions between di erent elements of the balances.Corrections were not made for base pressures.Balance-related calculated uncertainties in the measured static aerodynamic coe cients are given in table III.These uncertainties are based on balance output signals related to forces and moments by a laboratory calibration that is accurate to60.5per-cent of the rated load for each component.(See ta-ble I.)For the AFE,the moment reference center is4located at the center of the rake plane.(See g.4.) Thus,moments reduced about the model rake-plane center and reported herein have greater uncertainties than those measured at the balance moment center. The yawing and rolling moments at the balance have an uncertainty of only60.5percent of the rated load, whereas the moment at the rake-plane center also in-cludes uncertainties associated with the forces in the transfer equation.The transfer equation isYawing moment RP=Yawing moment B0(X)(Side force)andRolling moment RP=Rolling moment B0(Z)(Side force)where the subscripts RP and B denote the rake-plane center and the balance moment center,respectively. The transfer distances X and Z are de ned in g-ure4.In coe cient form,the uncertainty1related to the balance calibration for the side force is1C y=6(0:005)(Force rating)q1SThe uncertainty for the yawing moment is1C n;B=6(0:005)(Moment rating)q1dSand an identical equation applies for the rolling mo-ment.These balance uncertainties are su cient for measurements at the balance moment center.How-ever,at the rake-plane center,the yawing-moment uncertainty is1C n;RP=62401C n;B12+1C y X!2350:5and the rolling-moment uncertainty is1C l;RP=62401C l;B12+1C y Zd!2350:5Note that all the terms include the free-stream dy-namic pressure in the denominator so that the un-certainties are less at test conditions where q1is large|that is,at a higher Reynolds number rather than at a lower Reynolds number.The uncertainty in dynamic pressure is63percent.The ow condi-tions for which the present uncertainties have been calculated are presented in table II.Results and DiscussionsThe aerodynamic data from the Mach10air tests are tabulated in table IV.The Mach6results are presented in tables V and VI for air and in table VII for CF4.The test Reynolds number and model diameter are indicated in each table title.The aerodynamic coe cients C y,C n,and C l are plotted for an angle-of-sideslip range at various an-gles of attack in each facility and presented in g-ures5{7for Mach10in air,Mach6in air,and Mach6 in CF4,respectively.Data obtained at Mach6in air( g.6)indicated no e ect of Reynolds number on measured lateral and directional coe cients for a factor-of-4increase in postshock Reynolds num-ber.(Similar trends with respect to Reynolds num-ber were also observed for AFE longitudinal aero-dynamic characteristics presented in ref.16in which a negligible e ect of Reynolds number was noted for Mach6and10in air and at Mach6in CF4.) Therefore,the assumption is made that the e ect of Reynolds number on measured lateral and direc-tional data at Mach10in air and Mach6in CF4 is also negligible.The data are amenable to linear curve ts as shown in gures5{7,for which the ordi-nate scale is quite sensitive.These curves would be expected to go through the origin because the model was symmetrical about the pitch plane.However,as observed in gures5{7,an o set exists.This o set may be attributed to model misalignment or to any small stray signal in the data system that could cause a constant data o set because of the very small val-ues being measured relative to the load range of the balance.For example,if a slight misalignment of the model in the roll direction were introduced during model setup or if the balance location within the model were slightly misaligned,thereby producing a small o set in the center of gravity location(that is,within a few thousandths of an inch)in the side plane(y di-rection in g.4),then the e ect of the large axial-force component on this small moment arm may pro-duce a continuous bias in the measured quantities. For instance,from reference16at = =0 , Re1=0:462106/ft,and Mach6in CF4,the axial-force coe cient is1.382.The yawing-moment coe -cient,from table VII for similar conditions,is0.004. In much the same way as the change in the cen-ter of pressure in longitudinal aerodynamics is lo-cated,forming the ratio of yawing-moment coe -cient to axial-force coe cient yields the moment arm in the y direction,which for this case is approxi-mately0.003in.and thus within acceptable fabri-cation tolerances.A second linear curve,parallel to the data-faired curve,is drawn through the origin in5each part of gures 5{7.Values from measurements and the curve through the origin of gures 5{7are presented in tables IV{e of the slopes of these parallel curves through the origin to represent the lateral and directional stability derivatives should be valid because the data curves are linear through the test sideslip range.The lateral and directional stability derivatives are presented in gure 8and table VIII through the range of angle of attack for which tests were per-formed in each facility.For all test conditions,the con guration was laterally and directionally stable,as indicated by the positive values of C n and nega-tive values of C l .A comparison of lateral and direc-tional stability derivatives obtained at Mach num-bers of 6and 10in air illustrates no signi cant e ect of Mach number on stability characteristics ;a comparison of these stability derivatives with those obtained at Mach 6in CF 4indicates a small but measurable e ect of normal-shock density ratio on lateral and directional stability characteristics.Al-though the numerical values for air and CF 4are not greatly di erent,the data trends in air and CF 4ap-pear to be opposite.(Similar trends were observed in the longitudinal aerodynamic characteristics dis-cussed in ref.16.)This trend is most obvious for C l ,wherein the small numerical values require an expanded scale on the graph.The wind tunnel re-sults in CF 4are believed to be a better simulation of ight data than those in air because the shock de-tachment distance for CF 4is closer to the distance predicted for the actual ight case.(For example,see refs.6and 16.)Concluding RemarksStatic lateral and directional stability character-istics were obtained for the Aeroassist Flight Exper-iment (AFE)con guration through a range of angle of attack from 010 to 10 .Tests were conducted on two di erent-sized models at Mach numbers of 6and 10in air and at a Mach number of 6in tetra- uoromethane (CF 4).The e ects of Mach number,Reynolds number,and normal-shock density ratio on lateral and directional stability characteristics were examined.Changes in Mach number from 6to 10in air or in Reynolds number by a factor of 4at Mach 6had a negligible e ect on the lateral and directional sta-bility characteristics of the baseline AFE con gura-tion.Variations in density ratio across the normal portion of the bow shock from approximately 5(air)to 12(CF 4)had a measurable e ect on lateral and directional aerodynamic coe cients,but no signi -cant e ect on lateral and directional stability char-acteristics.The tests in air and CF 4indicated that the con guration is laterally and directionally stable through the test range of angle of attack as indicated by the positive values of C n and negative values of C l (positive e ective dihedral).In late 1991,the AFE program was cancelled and thus ended research e orts on this con guration.The realization of an AFE ight in the future is possible but uncertain.Hence,this paper documents the lateral and directional aerodynamic characteristics of the baseline AFE vehicle for use in the design of future aeroassist space transfer vehicles.NASA Langley Research Center Hampton,VA 23681-0001March 25,1993References1.Walberg,Gerald D.:A Review of Aeroassisted Orbit Transfer.AIAA-82-1378,Aug.1982.2.Wells,William L.:Wind-Tunnel Pre ight Test Program for Aeroassist Flight Experiment.Technical Papers|AIAA Atmospheric Flight Mechanics Conference ,Aug.1987,pp.151{163.(Available as AIAA-87-2367.)3.Wells,William L.:Free-Shear-Layer Turning Angle in Wake of Aeroassist Flight Experiment (AFE)Vehicle at Incidence in M =10Air and M =6CF4.NASA TM-100479,1988.4.Micol,John R.:Experimentaland Predicted Pressure and Heating Distributions for Aeroassist Flight Experiment Vehicle.J.Thermophys.&Heat Transf.,July{Sept.1991,pp.301{307.5.Wells,WilliamL.:SurfaceFlow and HeatingDistributions on a Cylinder in Near Wake of Aeroassist Flight Experi-ment (AFE)Con guration at Incidence in Mach 10Air.NASA TP-2954,1990.6.Micol,John R.:Simulation of Real-Gas E ects on Pres-sure Distributions for Aeroassist Flight Experiment Vehi-cle and Comparison With Prediction.NASA TP-3157,1992.7.Jones,Jim J.:The Rationale for an Aeroassist Flight Experiment.AIAA-87-1508,June 1987.8.Walberg,G.D.;Siemers,P.M.,III;Calloway,R.L.;and Jones,J.J.:The Aeroassist Flight Experiment.IAF Paper 87-197,Oct.1987.9.Gamble,Joe D.;Spratlin,Kenneth M.;and Skalecki,Lisa M.:Lateral Directional Requirements for a Low L/D Aeromaneuvering Orbital Transfer Vehicle.A Collection of Technical Papers|AIAA Atmospheric Flight Mechan-ics Conference,Aug.1984,pp.402{413.(Available as AIAA-84-2123.)610.Jones,Robert A.;and Hunt,James L.(appendix Aby James L.Hunt,Kathryn A.Smith,and Robert B.Reynolds and appendix B by James L.Hunt and Lillian R.Boney):Use of Tetra uoromethane To Simulate Real-Gas E ects on the Hypersonic Aero dynamics of Blunt Vehicles.NASA TR R-312,1969.11.Cheatwood,F.McNeil;DeJarnette,Fred R.;and Hamil-ton,H.Harris,II:Geometrical Description for a Pro-posed AeroassistedFlight ExperimentVehicle.NASA TM-87714,1986.ler, C.G.:Langley Hypersonic Aerodynamic/Aerothermodynamic Testing Capabilities|Present and Future.AIAA-90-1376,ler,Charles G.,III;and Gno o,Peter A.:PressureDistributions and Shock Shapes for12.84 /7 On-Axis and Bent-Nose Biconics in Air at Mach6.NASA TM-83222,1981.14.Midden,Raymond E.;and Miller,Charles G.,III:De-scription and Calibration of the Langley Hypersonic CF4 Tunnel|A Facility for Simulating Low Flow as Occurs for a Real Gas.NASA TP-2384,1985.15.Micol,John R.;Midden,Raymond E.;and Miller,CharlesG.,III:Langley20-Inch Hypersonic CF4Tunnel:A Facil-ity for Simulating Real-Gas E ects.AIAA-92-3939,July 1992.16.Wells,William L.:Measured and Predicted AerodynamicCoe cients and Shock Shapes for AeroassistFlight Exper-iment(AFE)Con guration.NASA TP-2956,1990.7。

两水平因子分析介绍本教程演示如何将Design Expert®软件用于两水平因子分析设计。

这些设计将帮助您筛选许多因子，以发现至关重要的少数，也包括他们可能存在的交互作用。

如果你赶时间，跳过“Note备注，笔记”这一部分，这种侧边栏，是提供给那些想花更多的时间来探索事物的人的。

备注程序的基本特性：在继续学习本教程之前，请返回到一个通用的单因子实验教程。

此处将不详细介绍演示的功能。

你现在要分析的数据来自道格拉斯·蒙哥马利的教科书《实验的设计与分析》，约翰·威利和儿子公司，纽约出版社。

本案例：华夫板（晶圆薄片）制造商必须立即降低用于现场过滤器加工助剂的甲醛浓度，否则将被监管官员关闭（安全法规强制要求）。

为了系统地探索他们的操作方法，工艺工程师对关键因子进行了全因子两水平设计，包括当前水平的浓度和可接受的低浓度。

全因子设计实例的因子和水平在这些过程的每一次组合中，实验者都记录下过滤速率。

我们的目标是最大限度地提高滤液过滤速率，并尝试找到条件，允许减少甲醛浓度，即因子C。

这个案例研究展示了设计专家提供的，许多两水平设计的特点。

这会让你在成为一名强大用户的道路上走得很好。

让我们继续！备注如果温度等因子很难改变，该怎么办：理想情况下，实验的运行顺序将完全随机化，这是设计专家默认的实验安排。

如果由于一个或多个因子很难被快速改变，而无法实现这一点，请选择裂区设计。

但是，请记住，对于随机性中受到限制的因子，您将付出降低效率标准的代价。

在开始一个裂区之前，参加“特性之旅”，了解设计专家是如何设计这样一个实验的，以及在选择效应时要注意什么，等等。

设计实验启动程序并单击“新建设计”。

你现在看到屏幕上有四个分支项。

继续使用因子Factorial因子选项，这是默认的。

您将使用默认的选择: Randomized Regular Two Factorial常规随机化两水平因子分析（析因）。

两水平析因设计生成器备注Design Expert的Design builder设计建模，提供2到21个因子的完整和部分两水平因子设计，以2的幂（4、8、16…）表示，最多可运行512次（实验组合）。

综合法和分析法一、教学目标：(一)知识与技能：结合已经学过的数学实例，了解直接证明的两种基本方法：分析法和综合法；了解分析法和综合法的思考过程、特点。

(二)过程与方法：培养学生的辨析能力和分析问题和解决问题的能力；(三)情感、态度与价值观：通过学生的参与，激发学生学习数学的兴趣。

二、教学重点：了解分析法和综合法的思考过程、特点三、教学难点：分析法和综合法的思考过程、特点四、教学过程：(一)导入新课：合情推理分归纳推理和类比推理，所得的结论的正确性是要证明的。

数学结论的正确性必须通过逻辑推理的方式加以证明。

本节我们将学习两类基本的证明方法：直接证明与间接证明。

(二)推进新课：1. 综合法在数学证明中，我们经常从已知条件和某些数学定义、公理、定理等出发，通过推理推导出所要的结论。

例如：已知a,b>0,求证a(b2c2) b(c2a2)亠4abc教师活动：给出以上问题，让学生思考应该如何证明，弓I导学生应用不等式证明。

教师最后归结证明方法。

学生活动：充分讨论，思考，找出以上问题的证明方法设计意图：弓I导学生应用不等式证明以上问题，弓I出综合法的定义证明：因为b2c2_ 2bc, a 0，所以a(b2c2) _ 2abc。

因为c2a2_ 2ac,b 0，所以b(c2a2) _ 2abc。

因此a(b2c2) b(c2a2)亠4abc。

一般地，禾U用已知条件和某些数学定义、公理、定理等，经过一系列的推理论证，最后推导出所要证明的结论成立，这种方法叫做综合法。

用P表示已知条件、已有的定义、定理、公理等,Q表示要证明的结论，则综合法可表示为：P= Q<—;(Qi= Q2)—：.Q2= Q3 l；..…—；：Qn 二Q综合法的特点是：由因导果，即由已知条件出发，利用已知的数学定理、性质和公式，推出结论的一种证明方法。

例1、在厶ABC中,三个内角A,B,C的对边分别为a,b,c,且A,B,C成等差数列，a,b,c成等比数列,求证△ ABC为等边三角形.分析：将A , B , C成等差数列，转化为符号语言就是2B =A + C; A , B , C ABC 的内角，这是一个隐含条件，明确表示出来是A + B + C =二；a , b，c成等比数列，转化为符号语言就是b2二ac •此时，如果能把角和边统一起来，那么就可以进一步寻找角和边之间的关系，进而判断三角形的形状，余弦定理正好满足要求•于是，可以用余弦定理为工具进行证明.证明：由A, B, C成等差数列，有2B=A + C . ①因为A,B,C ABC的内角，所以 A + B + C=二. ②由①②，得B==. ③3由a, b, c成等比数列，有b2二ac ④由余弦定理及③，可得2 2 2 2 2b ac -2accosB=a c -ac .再由④，得a2，c2-ac=ac.即(a -打=Q因此 a = c .从而A=C .由②③⑤，得71A=B=C= —.3所以△ ABC为等边三角形.注：解决数学问题时，往往要先作语言的转换，如把文字语言转换成符号语言，或把符号语言转换成图形语言等.还要通过细致的分析，把其中的隐含条件明确表示出来.例2、已知a,b R ,求证a a b b_a b b a.分析：本题可以尝试使用差值比较和商值比较两种方法进行。