微积分网课代修|微分学代写Differential calculus代考|MATH350 To the teacher

微积分网课代修|微分学代写Differential calculus代考|MATH350 To the teacher

简单的说,学好微积分(数学分析)是一个毁灭自己的先天直觉然后重新塑造一个后天直觉。

转变思维永远不是简单,但是不转变,贪图一时的捷径只是饮鸩止渴罢了。高中的时候,我一个同学很背单词的时候喜欢用汉字去拼那些单词的发音,还喜欢学各种解题技巧,这个时候我和他的成绩是一样的。

国外的老师较为看重学生homework的完成情况,对于同学们来说,完成一门科目作业并获得不错的成绩是尤为重要的事情。但对于不少同学来说,在自身英语说存在局限的情况下,当数学基础较为薄弱时,微积分作业的难度一下子就提升了,很难独立完成微积分作业。Calculus-do™提供的专业微积分代写能为大家解决所有的学术困扰,我们不仅会帮大家完成作业,还提供相应的数学知识辅导课程,以此来提高同学们学习能力。

我们提供的econ代写服务范围广, 其中包括但不限于:

  • 单变量微积分
  • 多变量微积分
  • 傅里叶级数
  • 黎曼积分
  • ODE
  • 微分学
微积分网课代修|微分学代写Differential calculus代考|MATH350 To the teacher

微积分网课代修|微分学代写Differential calculus代考|To the teacher

The bulk of the material in the book comes from my lecture notes.
There is little emphasis on closed-form computations and algebraic manipulations. I do think that a person who has never integrated by hand (or differentiated, or applied the quadratic formula, etc.) cannot possibly understand integration (or differentiation, or quadratic functions, etc.). However, a large proportion of time and effort can and should be directed toward:

  • understanding of the concepts and
  • modeling in realistic settings.
    The challenge of this approach is that it requires more abstraction rather than less.
    Visualization is the main tool used to deal with this challenge. Illustrations are provided for every concept. big or small. The pictures that come out are sometimes very precise but sometimes serve as mere metaphors for the concepts they illustrate. The hope is that they will serve as visual “anchors” in addition to the words and formulas.

It is unlikely that a person who has never plotted the graph of a function by hand can understand graphs or functions. However, what if we want to plot more than just a few points in order to visualize curves: surfaces, vector fields, etc.? Spreadsheets were chosen over graphic calculators for visualization purposes because they represent the shortest step away from pen and paper. Indeed, the data is plotted in the simplest manner possible: one cell – one number – one point on the graph. For more advanced tasks such as modeling, spreadsheets were chosen over other software and programming options for their wide availability and, above all, their simplicity. Nine out of ten, the spreadsheet shown was initially created from scratch in front of the students who were later able to follow my footsteps and create their own.

About the tests. The book isn’t designed to prepare the st udent for some preexisting exam; on the contrary; assignments should be based on what has been learned. The students’ understanding of the concepts needs to be tested but, most of the time, this can be done only indirectly. Therefore, a certain share of routine, mechanical problems is inevitable. Nonetheless, no topic deserves more attention just because it’s likely to be on the test.
If at all possible, don’t make the students memorize formulas.
In the order of topics, the main difference from a ty pical calculus textbook is that sequences come before ev erything else. The reasons are the following:

  • Sequences are the simplest kind of functions.
  • Limits of sequences are simpler than limits of general functions (including the ones at infinity).
  • The sigma notation, the Riemann sums, and the Riemann integral make more sense to a student with a solid background in sequences.
  • A quick transition from sequences to series often leads to confusion between the two.
  • Sequences are needed for modeling, which should start as early as possible.

微积分网课代修|微分学代写Differential calculus代考|From the discrete to the continuous

It’s no secret that a vast majority of calculus students will never use what they have learned. Poor career choices aside, a former calculus student is often unable to recognize the mathematics that is supposed to surround him. Why does this happen?

Calculus is the science of change. From the very beginning. its peculiar challenge has been to study and measure continuous change: curves and motion along curves. These curves and this motion are represented by formulas. Skilful manipulation of those formulas is what solves calculus problems. For over 300 years, this approach has been extremely successful in sciences and engineering. The successes are well-known: projectile motion, planetary motion, flow of liquids, heat transfer, wave propagation, etc. Teaching calculus follows this approach: An overwhelming majority of what the student does is manipulation of formulas on a piece of paper. But this means that all the problems the student faces were (or could have been) solved in the 18 th or 19 th centuries!

This isn’t good enough any more. What has changed since then? The computers have appeared, of course, and computers don’t manipulate formulas. They don’t help with solving – in the traditional sense of the word – those problems from the past centuries. Instead of continuous, computers excel at handling incremental processes, and instead of formulas they are great at managing discrete (digital) data. To utilize these advantages, scientists “discretize” the results of calculus and create algorithms that manipulate the digital data. The solutions are approximate but the applicability is unlimited. Since the 20 th century, this approach has been extremely successful in sciences and engineering: aerodynamics (airplane and car design), sound and image processing, space exploration, structure of the atom and the universe, etc. The approach is also circuitous: Every concept in calculus starts – of ten implicitly – as a discrete approximation of a continuous phenomenon!

微积分网课代修|微分学代写Differential calculus代考|MATH350 To the teacher

微积分网课代修|微分学代写Differential calculus代考|To the teacher

书中的大部分材料来自我的讲义。
很少强调封闭形式的计算和代数运算。我确实认为,一个从未手动积分(或微分,或应用二次公式等)的人不可能理解积分(或微分,或二次函数等)。然而,大部分时间和精力可以而且应该用于:

  • 对概念的理解和
  • 在现实环境中建模。
    这种方法的挑战在于它需要更多的抽象而不是更少的抽象。
    可视化是用来应对这一挑战的主要工具。为每个概念都提供了插图。大或小。出现的图片有时非常精确,但有时仅作为它们所说明概念的隐喻。希望它们除了单词和公式之外,还可以作为视觉“锚”。

从未手工绘制过函数图形的人不太可能理解图形或函数。但是,如果我们想要绘制的不仅仅是几个点以便可视化曲线怎么办:曲面、矢量场等?出于可视化目的,选择电子表格而不是图形计算器,因为它们代表了离笔和纸最短的一步。事实上,数据是以最简单的方式绘制的:一个单元格 – 一个数字 – 图表上的一个点。对于建模等更高级的任务,选择电子表格而不是其他软件和编程选项,因为它们具有广泛的可用性,最重要的是它们的简单性。十分之九,显示的电子表格最初是在学生面前从头开始创建的,这些学生后来能够跟随我的脚步并创建自己的。

关于测试。这本书不是为学生准备一些预先存在的考试而设计的。相反; 作业应该以所学内容为基础。需要测试学生对概念的理解,但大多数情况下,这只能间接进行。因此,有一定的套路,机械问题在所难免。尽管如此,没有一个话题仅仅因为它可能在测试中就值得更多关注。
如果可能的话,不要让学生记住公式。
在主题的顺序上,与典型的微积分教科书的主要区别在于序列排在其他所有内容之前。原因如下:

  • 序列是最简单的函数。
  • 序列的极限比一般函数的极限(包括无穷大的极限)更简单。
  • sigma 符号、黎曼和和黎曼积分对于具有扎实序列背景的学生来说更有意义。
  • 从序列到序列的快速过渡通常会导致两者之间的混淆。
  • 建模需要序列,应该尽早开始。

微积分网课代修|微分学代写Differential calculus代考|From the discrete to the continuous

绝大多数微积分学生永远不会使用他们所学的知识,这已不是什么秘密。除了糟糕的职业选择之外,以前的微积分学生通常无法识别应该围绕他的数学。为什么会这样?

微积分是变化的科学。从一开始就。它的特殊挑战是研究和测量连续变化:曲线和沿曲线的运动。这些曲线和这种运动由公式表示。巧妙地处理这些公式是解决微积分问题的方法。300 多年来,这种方法在科学和工程领域非常成功。成功是众所周知的:抛射运动、行星运动、液体流动、热传递、波传播等。微积分教学遵循这种方法:学生所做的绝大多数事情是在一张纸上操作公式。但这意味着学生面临的所有问题在 18 世纪或 19 世纪都已经(或可能已经)解决了!

这已经不够好了。从那以后发生了什么变化?当然,计算机已经出现,并且计算机不会操纵公式。它们无助于解决过去几个世纪的问题——传统意义上的问题。计算机不是连续的,而是擅长处理增量过程,而不是公式,它们擅长管理离散(数字)数据。为了利用这些优势,科学家们将微积分的结果“离散化”并创建操作数字数据的算法。解决方案是近似的,但适用性是无限的。自 20 世纪以来,这种方法在科学和工程领域非常成功:空气动力学(飞机和汽车设计)、声音和图像处理、太空探索、原子和宇宙结构等。

微积分网课代修|微分学代写Differential calculus代考|MATH350 To the teacher
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