* 1960s:
A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
* 1970s:
A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
* 1970s (new math):
A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
* 1980s:
A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word “potatoes” and discuss with your classmates.
* 1990s:
A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.
* 十七世纪 ： 
农民(peasant)的一袋土豆卖了10美元。他的成本为4/5的销售价格。他的利润是多少？ 
* 十八世纪前期： 
农民(farmer)的一袋土豆卖了10美元。他的成本为4/5的销售价格，也就是8美元。他的利润是多少？ 
* 十八世纪后期(新数学) ： 
一位农民用土豆组成的集合P换了钱组成的集合M。集合M的势等于10 ，M中每个元素的值为1美元 。画10个大圆点代表集合M中的元素。成本集合C也这样表示，它比M少两个大圆点。把C作为M的子集并回答如下问题：利润集合的势是多少？ 
* 20世纪80年代：
农民的一袋土豆卖了10美元。他的生产成本是8美元，他的利润是2美元。把“土豆”加上下划线 ，并与你的同学讨论。 

* 20世纪90年代： 
农民的一袋土豆卖了10美元。他或她的生产成本是0.8倍的他或她的收入。用你的计算器画出收入-成本图像。运行”马铃薯”程序，以确定利润。与你小组的同学讨论你的结果。写一篇简短的论文，在现实世界中的经济学中分析这个例子。 

来源：http://www.tvdsb.on.ca/banting/courses/math/Comics.htm 

当两个连续函数都趋于无穷时，我们常用洛必达法则来比较它们趋向无穷的快慢。函数的阶越高，它趋向无穷的速度就越快。在定义在正整数域上的函数中，n!趋向于正无穷的速度非常快，所以在算法设计中如果出现这样的时间复杂度就太糟糕了。logn趋向无穷的速度则非常慢。
今天你将看到一个增长速度比n!快的多的函数和一个比logn慢的多的函数，它们都源于一个函数–’Ackerman Function’。
并非一切的递归函数都有通项公式，Ackerman函数就是这么一个例子。它是一个双递归函数，有两个自变量(独立)。定义如下： 
 
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