Calculus, Corners, 6 min read

by Matthew Valverde
Calculus, Corners, and Salvation through Christ<span class="wtr-time-wrap after-title"><span class="wtr-time-number">6</span> min read</span>
Reading Time: 6 minutes


作者要感谢2024届毕业生(从左到右,作者在中间)奇迹·约翰逊, Gabe Rabold, Alex Hussar, and Jayden Yon for their assistance in writing this blog. 

In the fall of 2022, 微积分第一节课像往常一样开始了——学生和老师之间友好的对话, preparing for the pledges to the Bible and flags, and getting ready for the opening thoughts of the day.

这个特殊的课程为一周中的每一天创造了一个名字,目的是每天早上深入思考一些事情 Mull-It-Over Monday, Talk-About-It Tuesday, Witty Wednesday, Thoughtful Thursday, and Fink (think)-About-It Friday.  As the teacher, 我欣然接受这个机会,带领我的学生以一段经文来娱乐, a robust thought, or an interesting quote to start. 一个早晨经常浮现在我的脑海中,并促使我与你们分享我们讨论中令人难以置信的对话, the reader. 我发现这次谈话改变了我对微积分的理解, general mathematics, and the majesty and transcendence of God.

Planes, Points, and Curves Represent Reality

On this particular morning, 我们已经开始复习前一年的微积分原理,并开始讨论不同图的性质和特征. Here is a very shortened version of what we were analyzing. 如果你还不是一个数学家,我希望我能让你觉得自己是一个微积分新手.

你看,我们在微积分中会用到不同的数据集和方程. 这些信息可以在x-y坐标平面上进行转换和图形化表示. 点被绘制出来,曲线被绘制出来并连接起来,我们就有了对现实的视觉表现. 大多数曲线的外观要么向上移动,


the downward direction,





If the latter is true about the look of a curve on a graph, 图上至少有一个点存在从一个方向到另一个方向的过渡, either upward to downward or downward to upward.





If we look at the top of a mountain or the bottom of a valley, a horizontal line can be drawn to touch only that one point.


The horizontal line is neither increasing nor decreasing; there is no “slope” – the “slope” of that horizontal line is flat or zero. This is a point of transition, a point that is critical to the graph.

然而,有些图表没有明确的过渡点. 图形立即从向上转换到向下或从向下转换到向上. It is pointy or sharp, and it is an infinitely small point. We call this a corner. 角是两条曲线或直线的交汇处,一条增加,一条减少.


Deep Learning Results in Life Application

A student of mine, Alex, 我决定提出一个问题,这个问题让我比以往任何时候都更深入地思考微积分这个奇妙的学科. He asked, “Do corners exist in real life?” “Well, of course they exist,” I immediately thought to myself. Alex suggested that if we zoom in at an infinitely small level, 我们总能找到一个角不存在的点,那里会有一条曲线, or a point of transition, from decreasing to increasing or increasing to decreasing. Why? Because it is impossible to draw a perfect corner. 即使我们不受人为错误的限制,而是使用技术软件, we would still be limited by the constraints of the software, 因为计算机不能测量无限小的东西,最终会达到一个极限. 因此,人类不可能在物理世界或数字世界中创造一个真正的角落. Humans cannot create perfect corners because of earthly limitations. God, however, being outside of those physical and digital restrictions, is capable of creating corners. 最美丽的角落是在神的救恩故事中看到的,救恩是通过对基督耶稣的信仰而来的.

The Only True Corner Comes Through Christ

We know through reading the Scriptures that it is “by grace you have been saved through faith; and that not of yourselves, it is the gift of God” (Eph. 2:8, NASB1995). 我们知道救恩是唯一可能的,因为耶稣基督白白地给了我们恩典,因为父(使徒行传5:31)。. 我们也在福音书和使徒行传中读到耶稣行了许多神迹, 在那里,那些凭信心求医治的人立即得医治(马太福音. 8:1-4, 8:14-15; Acts 14:8-10). 这是一个清楚的证明,当一个人悔改的时候,救恩就立刻临到他, acknowledge what Jesus did for them on the cross, and receive forgiveness of their sins (Acts 16:31, Rom. 6:14) and freedom from condemnation (Rom. 8:1). One moment, the individual is destined to spend eternity in Hell away from God; in the next moment, because of God’s grace and mercy, 这个人现在注定要在天堂永远与神同在.

There is no middle ground. 没有与神分离或永远与神同在的中间阶段. Jesus tells us that “no one can serve two masters” (Matt. 6:24). In the same way, no one can be partially saved and partially condemned. We are either saved or condemned.

Thinking back to the mathematical example of the curve, if we graphed an individual’s salvation story, 在图形上绝不会有一个点的水平线与该点的“斜率”为零. It is an immediate change from death to life, from downward to upward. It is a corner! 这是我能想到的唯一一个在我们的世界里真正起作用的角落, 是上帝推翻了他所创造的世俗的限制,显示了他的能力和力量! 令我惊讶的是,微积分的一个基本概念(角)指向上帝的全能.

是上帝推翻了他所创造的世俗的限制,并显示了他的能力和力量. 微积分中的角的概念指出了上帝的全能. Click To Tweet

Salvation does not delay. 神恩典的大能,一旦由天父白白赐下,就不能被阻挡. Life change is immediate! I am thankful that God offers salvation to us at the corner, and not at the top of a mountain or the bottom of a valley. His love, grace, and mercy prompt immediate change. So, the next time that you see a mountain or a valley, or a graph of any kind in a mathematics course, textbook, or drawing, be thankful that God changes lives in a moment, and that calculus points back to the Savior.

If you are looking for a Christian school in Lancaster, PA, that places God’s Word at the center of every subject (even Calculus!) please schedule a private tour or register to attend one of our Open Houses. 如果你想了解更多BCK体育注册官网黎明基督教学院或 the Principle Approach, please call Karol Hasting at 717-285-2000.


Foundation Publications. (1995). New American Standard Bible.
Sullivan, Michael, and Kathleen Miranda. Calculus for the AP Course. 3rd ed., W. H. Freeman, 2020.