微积分入门指南：关键概念与应用详解

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《微积分简明介绍》是一本旨在为初学者和已有基础的学习者提供微积分基础知识概览的教材。本书内容覆盖了微积分的核心概念，从函数与图形的初步理解，到线性函数、极限、连续性等基础理论，再到导数及其应用，如链式法则、速度与变化率、增减函数和凹凸性分析。此外，书中还探讨了优化问题、指数与对数函数，以及求导和积分的概念，即反导数和积分方法。第1部分的章节包括： 1. 函数与图形：介绍了函数的基本概念，以及如何通过图形来理解和表达数学关系。 2. 线性函数、直线与线性方程：通过实例阐述了线性函数的特征及其在实际问题中的应用。 3. 极限：是微积分的基石，解释了当自变量接近某一值时函数行为的抽象概念。 4. 连续性：讨论了函数在某一点或区间上的连续性，这对于理解和计算导数至关重要。 5. 线性近似：通过线性函数来近似非线性函数的行为，有助于理解函数的变化趋势。 6. 导数的入门：导数是微分学的核心概念，它描述了函数在某一点处的变化率。 7. 微分法则：包括产品法则、商法则和链式法则，帮助学习者掌握求导的技巧。 8. 导数与速率：导数的应用之一，涉及速度、加速度等物理量的计算。 9. 增减函数：通过导数判断函数的增减性，理解函数的单调性。 10. 凸凹性：研究函数曲线的曲率，进一步深入理解函数的局部性质。 11. 优化问题：如何利用微积分解决实际生活中的最优化问题，如最小化或最大化某个量。 12. 指数与对数函数：这两种特殊类型的函数在微积分中有广泛应用，包括指数增长和对数模型。 13. 反导数与积分：导数的逆运算——不定积分，以及积分在物理、工程等多个领域的实际应用。第2部分的介绍部分强调，这些笔记并非详尽的数学教程，而是为了激发学习者运用微积分基本思想解决问题的能力，鼓励他们通过实际练习来掌握知识。本书还受益于国家科学基金会的资助，并声明了版权归属罗伯特·莫尔松。读者不能仅依赖笔记解答作业问题，而应该通过解决问题来深化理解和熟练运用微积分。《微积分简明介绍》是一本实用的学习指南，帮助读者快速理解微积分的基本原理，并鼓励他们在实践中巩固知识。无论是初学者还是需要复习的进阶者，都能从中找到适合自己的学习路径。

cost, c, and interest rate, r, is linear. You are given a cost rate table.

c r

300 7

400 8

Suppose you want to find c as a linear function of r. Write

c = Lr

= 300 + mr − 7

Now find m.

400 = 300 + m8 − 7 = 300 + m

100 = m

Hence

c = Lr = 300 + 100r − 7

Of course you can work this problem using the other methods for

finding the equation of a line, but it is easy to get things backwards

(usually the mistake that is made is getting the reciprocal of the slope).

We should finally mention one particular line that is not the graph of a

function. Recall again that a line is a geometric object. Suppose we

have drawn a vertical line in the x, y plane. Then there is no linear

function Lx such that the vertical line is the graph of the function Lx,

since a function Lx may take on at most one value for a given x,

However, it is possible to write an equation that represents the line. For

example, consider the vertical line through the point 2, 5. Since the

line is vertical, the x value of any point on the line must be 5, There is

no restriction on the y value of a point on the line. Hence the equation

of the line is

x = 2

Finally, we summarize three different forms for the equation of a line.

Each has an advantage in some setting.

y = a + mx − b, the equation of the line through a, b with slope m

y = c + mx, the equation of the line through 0, c with slope m

Ax + By + C = 0, the equation of a line that can be used to describe vertical lines

this with a calculator you will quickly come to the limiting value of

$1,105.20. In fact if you compounded 1000 times per year and then

2000 times per year the difference in the amount in the account at the

end of the year would be so small that it could not be indicated with the

number of digits available on the calculator.

Here is a second example of a problem that requires the concept of

limit to solve. Find the area of a circle of radius two (2. You might

object and say that every high school student knows the answer: π2

or just 4π. This is of course correct, but one might ask - what is the

decimal value of π. The answer to this question requires the concept of

limit.

Finally, here is a third example of the use of limit. In this case the

resulting limit is rather obvious, but the example still might help you

understand the concept. Suppose I start with $1000 in an account (

the account does not pay interest). I spend 1/2 of the money on 1

January. The next day I spend 1/2 of the remaining amount ($500).

The day after that, I spend 1/2 of the amount remaining. I continue this

process day after day. What is the limiting amount in my account? The

answer is clearly $0. It should also be clear that this will never happen!

Even after 10 years I would have some minuscule amount remaining in

the account (assuming we can subdivide currency as small as we like).

With these three examples, we are ready to consider an example using

functional notation. Suppose Fx = 1/x. Since this function is not

defined (does not make sense) if x = 0, we consider only positive

values of x. consider the following question. What is the limiting value

of Fx as x gets larger and larger. (You should see the connection with

the previous example). If we substitute larger and larger values for x

into the expression for Fx it becomes clear that the limiting value is 0.

We want to introduce some notation that will summarize the question

and the answer. First we need notation for the concept of “x gets larger

and larger”. You are probably already familiar with the notation that will

be used, but it can create a great deal of confusion. So be aware that

the concept of “infinity” is a very difficult one to define and understand.

The notation we use for “x gets larger and larger” is

x → ∞

and we say this as ”x tends to infinity”. Since we are considering larger

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微积分入门指南：关键概念与应用详解

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