The author wishes to thank Class of 2024 members (pictured from left to right with the author in the center) Miracle Johnsons, 加布Rabold, 亚历克斯轻骑兵, 还有Jayden Yon，感谢他们帮助我写这篇博客.
在2022年秋天, the calculus first-period class was beginning as usual – a friendly dialogue between students and teacher, 准备向圣经和国旗宣誓, 准备好迎接今天的开场思想.
This particular class coined a name for each day of the week with the goal of thinking deeply about something each morning – 仔细考虑过后的星期一, 周二谈论它, 诙谐的星期三, 深思熟虑的周四, 思考(思考)——星期五. 作为老师, I embrace the opportunity to lead my students in entertaining a portion of Scripture, 强大的思想, 或者用一句有趣的话来开头. One morning comes to my mind frequently and has prompted me to share this incredible dialogue from our discussion with you, 读者. I find that this conversation has transformed my understanding of calculus, 一般数学, 以及上帝的威严和卓越.
在这个特别的早晨, we had begun our review of calculus principles from the previous year and had begun to discuss the properties and characteristics of different graphs. 这是我们所分析的一个非常简短的版本. I hope that I will make you feel like a calculus novice by the end of this if you are not already a mathematician.
You see, there are different sets of data and equations that we deal with in calculus. This information can be translated and represented graphically on an x-y coordinate plane. Points are plotted, curves are drawn and connected, we have a visual representation of reality. The appearances of most curves will either move in the upward direction,
or there will be distinct sections of both upward and downward movement.
如果后者是正确的，那么图表上曲线的外观, there will be at least one point on the graph where there is a transition from one direction to the other, 不是从上到下就是从下到上.
When the curve transitions from upward to downward, we have reached the top of a mountain.
When the curve transitions from downward to upward, we have reached the bottom of a valley.
The horizontal line is neither increasing nor decreasing; there is no “slope” – the “slope” of that horizontal line is flat or zero. 这是一个过渡点，一个对图形至关重要的点.
However, there are some graphs where there is no definitive transition point. The graph instantaneously transitions from upward to downward or downward to upward. 它是尖的或锋利的，它是一个无限小的点. 我们称这个为角落. A corner is where two curves or lines, one increasing and one decreasing, meet.
我的一个学生, 亚历克斯, decided to pose a question that made me consider much more than ever before the wonderful subject of calculus. 他问道:“现实生活中存在角落吗?“好吧，它们当然存在，”我立刻对自己说. 亚历克斯建议如果我们放大到无限小的水平, we would always find a point where the corner does not exist and that there would be a curve, 或者是一个过渡点, 从减少到增加或从增加到减少. 为什么? 因为不可能画出一个完美的角. Even if we were not limited by human error and used technological software instead, 我们仍然会受到软件的限制, since computers cannot measure things infinitely small and eventually will reach a limit. Therefore, it is impossible for mankind to create a real corner in the physical or digital world. 由于地球的限制，人类无法创造出完美的角落. 神, 然而, 不受物理和数字的限制, 能创造角落吗. The most beautiful corner is seen in 神’s story of salvation that comes through faith in Christ Jesus.
We know through reading the Scriptures that it is “by grace you have been saved through faith; and that not of yourselves, 这是神的恩赐. 2:8 NASB1995). We know that salvation is only possible because of the grace Jesus Christ freely gives us because of the Father (Acts 5:31). We also read about many miracles Jesus carried out in the Gospels and Acts, where immediate healing was brought upon those requesting it by faith (Matt. 8:1-4, 8:14-15; Acts 14:8-10). This is a clear demonstration of the immediacy of salvation that is given to a person when they repent, 承认耶稣在十字架上为他们所做的一切, 并得赦免他们的罪(徒16:31), 罗. 6:14)和免于定罪的自由(罗马书6:14). 8:1). 一个时刻, the individual is destined to spend eternity in Hell away from 神; in the next moment, 因为神的恩典和怜悯, that same individual is now destined to spend eternity with 神 forever in Heaven.
没有中间地带. There is no in-between stage of being separated from 神 or being with 神 forever. 耶稣告诉我们，“一个人不能事奉两个主”(太六:1). 6:24). 同样，没有人可以部分得救，部分被定罪. 我们要么得救，要么被定罪.
回想一下曲线的数学例子, 如果我们画出一个人的救赎故事, there would never be a point on the graph where the “slope” of the horizontal line touching the point is zero. 这是从死到生，从下到上的直接变化. 它是一个角! It is the only corner that I can think of that truly works in our world, it is 神 who upends the laws of earthly limitations He created and displays His power and strength! It amazes me that one of the fundamental concepts of calculus (the corner) points to the omnipotence of 神.It is 神 who upends the laws of earthly limitations He created and displays His power and strength. The concept of the corner in calculus points to the omnipotence of 神. 点击推特
救赎不会拖延. The power of the grace of 神, once freely given by the Father, cannot be stopped. 生活变化是立竿见影的! 我感谢上帝在角落里为我们提供拯救, 而不是在山顶或谷底. 他的爱、恩典和怜悯使人立刻改变. So, 下次你看到山或山谷的时候, 或者是数学课上的任何图形, 教科书, 或画, 感谢上帝在一瞬间改变了生命, 而这个计算指向了救世主.
基金会的出版物. (1995). 新美国标准圣经.
迈克尔·沙利文和凯瑟琳·米兰达. AP课程的微积分. 3日艾德., W. H. 弗里曼,2020.