Calculus I: How Instant Speed Became Measurable
Limits, derivatives and integrals through the ideas that surprise people most, from 0.999... equaling 1 to why a can's best shape is as tall as it is wide.
Try 3 questions
1. The graph of f(x) = x² is redrawn as f(x − 3). Where does its vertex move?
- Three units to the right
- Three units downward
- Three units to the left
Show the answer
Answer: Three units to the right
Replacing x with x − 3 means the same output now happens 3 units later, so the whole graph slides right, not left.
2. Why does f(x) = x² have no inverse function over all real numbers?
- Two inputs give each positive output
- It grows too quickly to be reversed
- It is undefined for negative inputs
Show the answer
Answer: Two inputs give each positive output
Both 2 and −2 give 4, so an inverse wouldn't know which to return; restricting to x ≥ 0 fixes this and gives the square root.
3. For very large x, which grows faster: x¹⁰⁰ or 2ˣ?
- x¹⁰⁰, at every value of x
- 2ˣ, eventually and overwhelmingly
- Both grow at the same rate
Show the answer
Answer: 2ˣ, eventually and overwhelmingly
x¹⁰⁰ leads for a long while, but any exponential with base above 1 eventually outruns every polynomial, by a margin that keeps widening.
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