Microsoft Math支持下函数教学的探究精品文档7页

math.h常用函数-回复什么是math.h？在C语言中，math.h是一个常用的库文件，它包含了很多数学函数的定义和声明。

通过引入math.h文件，我们可以使用这些函数来进行各种数学运算。

为什么要使用math.h？在编程过程中，我们经常需要进行各种数学运算，比如求平方根、取绝对值、进行三角函数运算等等。

这些运算不仅可以在通常的数学计算中使用，也可以用于解决实际问题和算法的设计。

而math.h库文件提供了这些数学函数的实现，使得我们能够更方便地进行数学运算。

常用的math.h函数有哪些？1. abs函数：用于求取一个整数的绝对值。

其定义如下：int abs(int x);其中x为待求取绝对值的整数。

返回值为x的绝对值。

2. sqrt函数：用于求取一个数的平方根。

其定义如下：double sqrt(double x);其中x为待求取平方根的数。

返回值为x的平方根。

3. pow函数：用于求取某个数的幂。

其定义如下：double pow(double x, double y);其中x为底数，y为指数。

返回值为x的y次幂。

4. sin函数：用于求取一个角的正弦值。

其定义如下：double sin(double x);其中x为角度值（以弧度为单位）。

返回值为x的正弦值。

5. cos函数：用于求取一个角的余弦值。

其定义如下：double cos(double x);其中x为角度值（以弧度为单位）。

返回值为x的余弦值。

6. tan函数：用于求取一个角的正切值。

其定义如下：double tan(double x);其中x为角度值（以弧度为单位）。

返回值为x的正切值。

这些只是math.h库文件中的部分函数，在实际使用时，根据需要可能会使用到更多的数学函数。

这些函数涵盖了基本的数学运算，可以帮助我们完成一些常见的数学计算任务。

现在，让我们逐步来解释每个函数的用途并举例说明。

首先是abs函数，它用于取一个整数的绝对值。

math.h常用函数-回复"math.h常用函数"的主题涉及C语言数学库中的常见函数和它们的功能以及用法。

在本文中，将逐步回答与该主题相关的问题，并解释每个函数的作用。

第一部分：介绍math.h库和常见函数1. math.h是什么？math.h是C语言标准库之一，提供了许多数学运算的函数。

2. math.h库的包含和函数调用方法为了使用math.h库中的函数，我们需要在程序中包含头文件。

c#include <math.h>这样，我们就可以使用库中的函数。

3. 常见的math.h函数有哪些？math.h库提供了许多函数，用于数学运算、三角函数、取整和幂运算等。

其中一些常见的函数包括：- sin(x)：计算x的正弦值。

- cos(x)：计算x的余弦值。

- tan(x)：计算x的正切值。

- sqrt(x)：计算x的平方根。

- pow(x, y)：计算x的y次幂。

- ceil(x)：返回不小于x的最小整数值。

- floor(x)：返回不大于x的最大整数值。

- abs(x)：返回x的绝对值。

- log(x)：计算x的自然对数。

第二部分：详细介绍常用函数和它们的功能及用法1. sin(x)函数- 功能：计算x的正弦值。

- 用法示例：cdouble result = sin(0.5);这将计算0.5的正弦值，并将结果存储在result变量中。

2. cos(x)函数- 功能：计算x的余弦值。

- 用法示例：cdouble result = cos(1.2);这将计算1.2的余弦值，并将结果存储在result变量中。

3. tan(x)函数- 功能：计算x的正切值。

- 用法示例：cdouble result = tan(0.8);这将计算0.8的正切值，并将结果存储在result变量中。

4. sqrt(x)函数- 功能：计算x的平方根。

- 用法示例：cdouble result = sqrt(16.0);这将计算16.0的平方根，并将结果存储在result变量中。

5 信息技术支持的函数研究-北师大版高中数学必修第一册（2019版）教案教学目标•了解信息技术对于函数研究的支持作用•掌握在信息技术支持下使用函数进行实际问题建模和解决问题的方法•培养学生信息技术素养和创新意识教学重点•理解信息技术对函数研究的推动作用•掌握使用函数解决问题的方法•培养学生信息技术素养和创新思维教学难点•学生掌握信息技术与函数研究的结合方式•学生能否在实际问题中准确、合理地运用函数进行建模教学内容本节课将围绕信息技术支持的函数研究展开，包括以下内容：1.信息技术对函数研究的支持作用介绍信息技术在函数研究中的应用场景，例如使用电子表格软件绘制函数图像、编写程序模拟函数运算、使用数据分析软件处理实验数据等。

2.使用函数进行实际问题建模介绍如何将实际问题转化为数学模型，进而使用函数进行建模和求解。

例如，通过函数建模解决考试成绩排名问题、使用函数模拟赛车行驶等。

3.使用信息技术支持的函数研究解决问题通过具体案例展示如何运用信息技术支持下的函数研究解决实际问题，例如使用电子表格软件绘制收入支出曲线、使用数据分析软件处理环境数据等。

4.信息技术素养的培养通过课堂活动和案例演示，培养学生信息技术素养，提高创新思维能力。

教学方法本节课使用讲授和案例分析相结合的教学方法。

1.讲授通过PPT和讲述，介绍信息技术支持的函数研究的相关概念和技术，注意理论和实践相结合。

2.案例分析通过实际案例，演示如何使用函数进行建模和解决实际问题，提高学生的动手能力，既培养学生创新思维，又提高学生信息技术素养。

教学评价本节课的评价方式主要有以下几种：1.准确度评价使用学生提交的数学作业进行评价，评估学生的数学能力和实际问题解决能力。

2.创新力评价通过学生的课堂互动和实践表现评估学生的创新意识和动手能力。

3.信息技术素养评价考察学生信息技术素养的培养情况，例如对信息技术的认知、信息技术应用技巧和信息技术解决课堂问题的能力评价。

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。

第1章Microsoft Math简介本章初步介绍Microsoft Math(微软数学)的特点、基本功能、操作界面及基本使用常识。

通过对本章的学习，使读者能够初步应用Microsoft Math解决数学、科学计算问题，了解Microsoft Math是一款操作极其简便、功能非常强大，且覆盖学生基础课程的专业数学学习软件。

本章要点：●初览Microsoft Math强大功能。

●关于Microsoft Math操作界面。

●关于数学表达式与计算赋值。

●向导式解答及提供相关计算。

●方便快捷的手写输入方式。

●函数。

●解题示例。

●绘制图形。

●中断计算。

●关于大、小写。

●算子(运算符)的优先级。

●系统要求。

1.1 初览Microsoft Math的强大功能本节将重点介绍Microsoft Math的强大功能，使读者对其有一个基本的认识。

Microsoft Math的强大功能主要体现在7个方面：①强大的多种解方程、不等式或方程组的功能；②常用数学与科学公式和方程库(Formulas and Equations)；③向导式解答及提供相关计算；④直观形象的Graphing(图形计算器)；⑤三角形计算器；⑥单位转换器；⑦手写输入，方便快捷。

Microsoft Math能完成多种计算，其功能非常强大，是用户学习和工作的好帮手。

1.1.1 Microsoft Math的功用Microsoft Math尤其适合于学生和教师，可以帮助他们逐步解方程，更好地理解代数学(Algebra)、几何学(Geometry)、三角函数(Trigonometry)、物理学(Physics)、化学(Chemistry)、幂函数(Laws of Exponents)、对数函数(Properties of Logarithms)、微积分(Calculus)和各种常量(Constants)等，其基本功能如下。

(1) 数值计算。

Microsoft Math软件能够进行有理数、无理数计算，能够进行实数、复数计算，能够进行精确或近似计算，能够由用户设定计算精度。

北师大版九年级数学下册：第二章 2.3.2《确定二次函数的表达式》精品教案一. 教材分析北师大版九年级数学下册第二章第三节《确定二次函数的表达式》的内容是在学生已经掌握了二次函数的一般形式和图象的基础上进行讲解的。

本节课的主要目的是让学生学会如何根据二次函数的图象或者给定的条件来确定二次函数的表达式。

内容主要包括：待定系数法求二次函数的表达式，根据图象确定二次函数的顶点式，利用配方法将一般式化为顶点式。

这些内容对于学生来说，既有挑战性，又有实用性，对于提高学生的数学素养和解决实际问题的能力具有重要意义。

二. 学情分析学生在学习本节课之前，已经掌握了二次函数的一般形式和图象，对于如何从图象或给定条件中获取函数信息有一定的了解。

但是，对于如何运用待定系数法求解二次函数的表达式，如何根据图象确定二次函数的顶点式，以及如何利用配方法将一般式化为顶点式，可能还存在一定的困难。

因此，在教学过程中，需要教师引导学生通过观察、思考、操作、交流等活动，逐步掌握这些方法。

三. 教学目标1.让学生掌握待定系数法求解二次函数的表达式。

2.让学生学会如何根据二次函数的图象确定其顶点式。

3.让学生掌握利用配方法将二次函数的一般式化为顶点式。

4.培养学生的观察能力、思考能力、操作能力和交流能力。

四. 教学重难点1.教学重点：待定系数法求解二次函数的表达式，根据图象确定二次函数的顶点式，利用配方法将一般式化为顶点式。

2.教学难点：待定系数法求解二次函数的表达式，利用配方法将一般式化为顶点式。

五. 教学方法采用问题驱动法、案例教学法、小组合作学习法等，引导学生通过观察、思考、操作、交流等活动，掌握本节课的内容。

六. 教学准备1.准备相关的教学案例和图象。

2.准备教学PPT。

3.准备练习题。

七. 教学过程1.导入（5分钟）通过展示一些二次函数的图象，让学生观察并思考：这些图象有什么特点？你能从中获取哪些信息？从而引出本节课的主题——如何确定二次函数的表达式。