78数学建模

数学建模数学建模数学建模目录摘要························································································ⅠAbstract···················································································Ⅱ第1章绪论·············································································11.1数学建模的起源·····························································11.2数学建模的意义·····························································11.2数学建模的过程·····························································1第2章商品的最佳销售价格························································22.1问题重述······································································22.2模型假设与符号说明·······················································22.2.1模型的假设··························································22.2.2模型的符号说明····················································22.3问题的分析明································································22.3.1问题一································································22.3.2问题二································································22.4问题的解答···································································32.5章节小结······································································3第3章儿童问题行为与母亲不耐心程度关系研究·····························43.1问题重述······································································43.1.1问题背景·····························································43.2需要解决的问题·····························································43.2.1问题一································································43.2.2问题二································································43.2.3问题三································································43.2.4文明四································································43.3模型假设与符号说明·······················································43.3.1模型的假设··························································43.3.2模型的符号说明····················································53.4模型分析解答································································53.4.1模型的分析··························································53.4.2模型的解答··························································53.5章节小结······································································8第4章数值分析·······································································94.1问题重述······································································9数学建模4.2问题分析解答································································94.2.1插值多项式··························································94.3章节小结····································································12结论························································································13参考文献·················································································14数学建模I摘要本文第一个问题主要研究如何确定能有最大收益的销售价格问题，需要明确商品销售价格与之销售价格的关系，在研究过程中运用了一元二次函数以及求一元二次函数的导数的方法。第二个问题主要通过MATLAB进行数据的处理，回归方程系数的求解，回归方程显著性的检验，进行点估计，本文使用的方法可移植性强且MATLAB使用的语句都正文里出现。第三个问题运用了Lagrange公式求解)(xfy的插值多项式，Lagrange多项式和分段三次插值多项式。关键词函数的最值问题；MATLAB绘图；Lagrange公式；插值多项式数学建模IIAbstractThispapermainlystudieshowtodeterminethefirstproblemtohavethemaximumbenefitsofthesalesprice,salespricetoclearthesalepriceofthecommodityand,inthecourseofthestudyusingaquadraticfunctionandelementderivativemethodformonadicquadraticfunction.ThesecondmainproblemsofdataprocessingbyMATLAB,solvingthecoefficientsofregressionequation,regressionequationsaresignificant,forpointestimation,thispaperusethemethodofportabilityanduseofMATLABstatementsinthetextappear.ThethirdproblemusestheLagrangeformulatosolve)(xfyinterpolationpolynomial,Lagrangepolynomialandpiecewisethreeinterpolationpolynomial.KeywordsThemostvaluableproblemoffunction;MATLABdrawing;Lagrangeformula;interpolationpolynomial数学建模——绪论-1-第1章绪论1.1数学建模的起源数学建模是在20世纪60、70年代进入一些西方国家大学的，我国的几所大学也在80年代初将数学建模引入课堂。经过30多年的发展，现在绝大多数本科院校和许多专科学校都开设了各种形式的数学建模课程和讲座，为培养学生利用数学方法分析、解决实际问题的能力开辟了一条有效的途径。1.2数学建模的意义培养创新意识和创造能力；训练快速获取信息和资料的能力；锻炼快速了解和掌握新知识的技能；培养团队合作意识和团队合作精神；增强写作技能和排版技术；荣获国家级奖励有利于保送研究生；荣获国际级奖励有利于申请出国留学；更重要的是训练人的逻辑思维和开放性思考方式1.3数学建模的方法数学建模是一种数学的思考方法，是运用数学的语言和方法，通过抽象、简化建立能近似刻画并“解决”实际问题的一种强有力的数学手段。以下我们将通过模型准备、模型假设、模型建立、模型求解、模型分析、模型检验、模型应用的方法，利用数理统计、最优化、图论、二次方程、计算方法、层次分析法。模糊数学、MATLAB等手段，商品的最佳销售价格、儿童问题行为与母亲不耐心程度关系研究、解决数值分析三个问题。数学建模——商品的最佳销售价格-2-第2章商品的最佳销售价格2.1问题重述某商店以每件10元的进价购进一批衬衫，并设此种商品的需求函数2P-80=Q（其,Q为需求量，单位为件；P为销售价格，单位为元）。问该商品应将销售价定为多少元卖出，才能获得最大利润？最大利润为多少？2.2模型假设与符号说明2.2.1模型的假设2.2.1.1假设1本文题干提出的需求函数有效。2.2.1.2假设2销售价格为正数。2.2.1.3假设3需求量为正整数。2.2.2模型的符号说明表1-1符号说明符号符号说明Q该商品的需求量，单位为（个）P销售价格，单位为（元）W(P)总的利润，单位为（元）W′(P)W关于P的导数2.3问题的分析2.3.1问题一该商品不需要盈利，则其价格应定在}10{}40|{}100|{PPPP元的区间上,当价格定在商品价格}100|{PP价格每提高一些，亏损将会减少；价格定在}10{}40|{PP元时没有盈利也没有亏损，其中售价为}40|{PP元时，商品积压也是一种变相的损失。2.3.2问题二数学建模——商品的最佳销售价格-3-该商品若要盈利，则其价格应该定在}100|{PP元的区间上，且在该区间存在最大收益。2.4问题的解答根据题设有)280)(10()10()(PPQPPW(1.1)整理得8001002)(2PPPW(1.2)欲求W的最大值，则求W关于P的导数1004)(PPW(1.3)0)(PW令得唯一驻点25P显然25P是极大值点，也是最大值点。所以，有最大利润450)(PWMAX元。通过MATLAB进行公式（1.2）的绘制如图1-1所示图1-13.