I introduced differentials the past two years with the idea of predicting a stock price in the future. I'm not sure this was the best situation to use it in, but using that totally sensible anchor my students seem to understand the idea of differentials really well. Basically f(x+dx) = f(x) + f'(x)dx translates to “the population in the future will be approximately equal to the population now plus how fast the population is changing now times how many years in the future.” I think that putting this sentence with the equation for differentials is really helpful to make sense of it.

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