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What about being convinced that local extrema of smooth functions occur when the derivative is zero? You find it easier to work through an epsilon-delta algebra
by jordigh 10y ago
What about being convinced that local extrema of smooth functions occur when the derivative is zero? You find it easier to work through an epsilon-delta algebraic proof than to look at a graph and see that it's flat wherever there's a local extremum? Do you find graphs to represent data useless and you would rather look at long lists of numbers?
- wongarsu 10y agoThe derivative is the rate of change. If I chose any two points to the left and right to the local extremum, they are non-zero and have different signs. Since there has to be a zero between those two points, the local extremum must have a derivative of zero. That's the line of reasoning I would go. Looking at a graph, searching for exceptions and concluding that there aren't any is probably as easy in this case. Like probably most people I have a harder time thinking about epsilon-delta proofs. >Do you find graphs to represent data useless and you would rather look at long lists of numbers? No, graphs are great. They sacrifice precision to show simple relations between large amounts of numbers, and are really good at that. Numbers a row of numbers can't do that (the closest equivalent to a line graph is to replace each but the first number with their difference to the previous number, but that's still inferior). Natural language is lacking precision at reasonable density for the job. So graphs are pretty much the only decent tool for their job we have invented so far.