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

• 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|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|To the teacher

• 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”。

• 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。