By P. Fitzpatrick
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Additional resources for Advanced Calculus [Int'l Student edn]
An infinite sequence can point to the same result we’d find if we took it all at once. In calculus, there are detailed rules about how to find what result an infinite set of steps points to. And, there are certain sequences that cannot be worked out. But, for this primer, we’ll deal with functions that behave nicely. We’re used to jumping between finite representations of the same idea (5 = V = |||||). 333 . . 003 + . .. When we turned a circle into a ring-triangle, we said “The infinitely-many rings in our circle can be turned into the infinitely-many boards that make up a triangle.
He saw order after order go by without noticing the deeper pattern. The crime boss is different: he wants the indefinite integral, the pattern you are following. He’s savvy enough to track the pattern as the orders come in: “The area is increasing 1, 3, 5, 7. . ” Now, there are likely many shapes that could grow their area by 2x +1. But, combined with a constant perimeter increase of 4, he suspects you’re making a square garden after a few deliveries. How does the godfather do it? Again, by working backwards.
After all, they start 10 miles apart, and the gap shrinks at 10 miles per hour (5mph + 5mph). Therefore, the mosquito must be flying for an hour, and go 20 miles. Whoa! Did we just find the outcome of a process with an infinite number of steps? I think so! 40 CHAPTER 9. WORKING WITH INFINITY 41 Splitting A Whole Into Infinite Parts It’s time to turn our step-by-step thinking into overdrive. Can we think about a finite shape being split into infinite parts? • In the beginning of the course, we saw a circle could be split into rings.
Advanced Calculus [Int'l Student edn] by P. Fitzpatrick