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上海市精品课程配套教材·微积分基础:引入 Mathematica 软件求解(附光盘1张)

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上海市精品课程配套教材·微积分基础:引入 Mathematica 软件求解(附光盘1张)

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微积分是人类文明发展史上理性智慧的精华,它的出现,不仅彻底更新了数学的面貌,而且显著地促进了整个科学技术的发展。目前,微积分的理论与方法已广泛地应用于自然科学、工程技术乃至社会科学的各个领域。它提供给人们的不仅是一种高级的数学技术,而且是一种人类进步必需的文化素质和能力。
本书力求运用通俗的语言向读者介绍高等数学中最基础的知识,全书以微积分为核心,在高等数学学习中结合使用数学软件,通过参与 “演示与实验”帮助学生理解数学中的一些抽象概念和理论,并方便、简捷地用计算机来解复杂的实际运算问题。
Content Description

《微积分基础:引入Mathematica软件求解》以微积分为核心,在高等数学学习中结合使用数学软件,通过参与“演示与实验”帮助学生理解数学中的一些抽象概念和理论,并方便、简捷地用计算机来解复杂的实际运算问题。《微积分基础:引入Mathematica软件求解》引入国外先进的教学模式和教学理念,注重知识的实用性、生动性和趣味性,化解了过难过繁的运算技巧,将学生从枯燥的公式和大量的运算中解放出来。
Catalogue

第一章 数学与计算机
第一节 计算机与数学的关系
一、计算、计算方法和计算工具
二、计算机数学软件
三、Mathematica的特点
第二节 初等数学的计算机算法
一、Mathematica的启动和运行
二、用Mathematica作算术运算
三、用Mathematica作代数运算
四、用Mathematica作函数运算
五、用Mathematica解方程
六、用Mathematica作图
第二章 极限与连续
第一节 数列的极限
一、数列的概念
二、数列的极限
第二节 函数的极限
一、函数极限的定义
二、函数极限的性质
三、函数极限的基本运算
第三节 利用Mathematica计算极限
第四节 函数的连续性
一、f(x)在点x0的连续
二、间断点的类型
三、f(x)在区间上的连续性
第三章 一元函数微分学
第一节 导数的概念
一、导数概念实例
二、函数的变化率——导数
三、求函数y=f(x)的变化率(导数)的方法
四、可导与连续的关系
五、导数的几何意义
第二节 导数的运算
一、用导数的定义求导
二、导数基本运算法则和基本初等函数导数公式
三、反函数的导数
四、基本初等函数导数公式
五、复合函数的导数
六、利用Mathematica求导数
第三节 隐函数和参数方程所确定的函数的导数
一、隐函数的导数
二、参数方程所确定的函数的导数
第四节 高阶导数
一、高阶导数的概念
二、高阶导数的求导法则
三、利用Mathematica求高阶导数
第五节 函数的微分
一、微分的定义
二、可导与微分的关系
三、微分的几何意义
四、微分的运算法则
五、微分在近似计算中的应用
六、利用Mathematica求微分
第四章 导数的应用
第一节 利用导数求极限
一、中值定理简介
二、洛比达法则
第二节 函数的单调性
第三节 函数的极值与最值
一、函数的极值
二、函数的最大值与最小值
第四节 导数在经济分析中的应用
一、经济学中几个常用函数
二、边际函数
第五节 曲线的凹凸性
第六节 导数应用的Mathematica求解
第五章 不定积分和定积分
第一节 不定积分
一、不定积分的概念
二、不定积分的基本公式
三、不定积分的性质
四、基本积分方法
五、利用Mathematica计算不定积分
第二节 定积分
一、定积分的概念
二、定积分的性质
三、微积分的基本是理
四、利用Mathematica计算定积分
第三节 广义积分
一、无穷区间上的广义积分
二、无界函数的广义积分
第六章 定积分的应用
第一节 定积分在几何上的应用
一、利用定积分求平面图形的面积
二、利用定积分求体积
三、利用定积分求平面曲线的弧长
第二节 定积分在物理上的应用
一、变速直线运动的路程
二、变力沿直线所作的功
三、静止液体的压力
四、在电学上的应用
第三节 定积分在经济上的应用
第七章 常微分方程
第一节 微分方程的基本概念
一、微分方程的发展
二、微分方程的基本概念
第二节 如何建立微分方程
第三节 微分方程的求解
一、可分离变量的微分方程
二、一阶线性微分方程
三、二阶常系数线性微分方程
四、可降阶的高阶微分方程
第四节 利用Mathematica求解微分方程
一、可以准确求解的微分方程
二、微分方程(组)的数值解
第八章 无穷级数
第一节 无穷级数的概念
一、常数项无穷级数和函数项无穷级数
二、无穷级数的敛散性
三、利用Mathematica软件来判断级数的敛散性
第二节 无穷级数的性质与敛散性
第三节 正项级数
第四节 交错级数与任意项级数
一、交错级数
二、绝对收敛与条件收敛
第五节 幂级数
一、幂级数的收敛区间
二、幂级数的性质
第六节 幂级数在函数逼近中的应用
一、泰勒公式
二、泰勒级数
三、幂级数在近似计算中的应用
第九章 Mathematica系统提高篇
第一节 表和表的使用
第二节 平面图形的绘制
一、含参数的一元函数图形的绘制
二、一元隐函数图形的绘制
第三节 空间图形的绘制
一、空间曲面的绘制
二、空间曲线的绘制
三、绘制空间曲面的平面截线
四、绘制空间曲面的平面截线族
五、根据曲面网格点绘制曲面
六、利用图形考察多元函数的极值和最值
第四节 绘制微分方程的积分曲线
一、绘制微分方程的特解的积分曲线
二、绘制微分方程的通解的积分曲线族
三、绘制微分方程组的特解的相平面曲线
第五节 优化问题
第六节 插值与拟合
一、插值问题
二、拟合问题
第七节 幂级数与函数逼近
第八节 迭代算法
习题
附录一 Mathematica软件常用操作命令
附录二 微积分基本公式
附录三 初等数学部分公式
附录四 习题参考答案
后记
Book Abstract

工具的发明和改进,更好地体现算理的要求;但它反过来可以促进数学的发展,使其内容、方法更丰富,理论更完善,甚至促进数学在新的领域里再充实提高。事实上,历史上各种计算工具的演变,一方面是体现着如何更好地使数学的算理具体化和可操作化的过程;另一方面也是由于社会生产、发展而带来的要求计算工具不断提高其效能的过程。能体现这两个要求的计算工具才是有生命力的,反之必然被淘汰。
数学以适应计算工具特点的机械化过程是通过算法来表达的。算法是算理和计算工具之间的桥梁,或者是相互关系的综合体现。事实上,没有不具备算法的计算工具,也不存在不适应计算工具的算法。数学应该适当地改革自身以适应计算工具的特点,计算工具在数学中占有不可或缺的地位,起着特殊的作用。计算工具对数学发展的巨大影响,也就是计算对数学发展的促进作用。从这个意义上说,数学计算具有如下重要的意义。
(1)计算推动了应用数学的发展。
(2)计算加快了科学的数学化。
(3)计算促进了纯数学的发展。
算法是由一系列有限的规则所组成的一个过程。一个算法实质上就是解决一类问题的一个处方,它包括一套指令,只要一步一步地按照指令进行操作,就能引导到问题的解决。
Introduction

21世纪是一个变幻莫测、催人奋进的时代,科学技术飞速发展,知识更新日新月异,希望、困惑、挑战随时都有可能出现在每一个社会成员的生活之中。20世纪杰出的数学家冯·诺伊曼(Johny on Neumann,1903-1957)曾指出:“微积分是近代数学中最伟大的成就,对它的重要性无论作怎样的估计都不会过分。”同时,他又被称为计算机之父,他发明的“流程图”沟通了数学语言与计算机语言,因而诞生了世界上第一台电子计算机,开创了数学与计算机发展的新时代。
微积分是人类文明发展史上理性智慧的精华,它的出现,不仅彻底更新了数学的面貌,而且也促进了整个科学技术的发展,目前,微积分的理论与方法已广泛地应用于自然科学、工程技术乃至社会科学等各个领域,它提供给人们的不仅是一种高级的数学技术,而且是一种人类进步所必需的文化素质和能力。
诚然,学习和掌握一定程度的微积分知识,不仅是对理工类学生的要求,而且也是对经济管理、人文科学等各类学生的基本要求和必备要素,但是由于数学的抽象表达和符号语言与人们的实际生活距离较大,给微积分的教与学带来了很大的障碍和困难,因此在大学的“微积分”教学过程中仍然存在许多不尽如人意的地方:抽象难教,枯燥难学,空洞难用,以致使本来生动实用的一门课程成为学校中老师与学生的难点;以致使微积分这一近代数学中最伟大的成就,难以为人们所普及、运用,难以更好地服务于社会。

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