Lesson plan
Increment theorem
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function is differentiable at and that is infinitesimal. Then \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x for some infinitesimal , where \Delta y=f(x+\Delta x)-f(x). If \Delta x \neq 0 then we may write \frac{\Delta y}{\Delta x} = f'(x) + \varepsilon, which implies that \frac{\Delta y}{\Delta x}\approx f'(x) , or in other words that \frac{\Delta y}{\Delta x} is infinitely close to f'(x) , or f'(x) is the standard part of \frac{\Delta y}{\Delta x} . A similar theorem exists in standard Calculus. Again assume that is differentiable, but now let be a nonzero standard real number. Then the same equation \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x holds with the same definition of , but instead of being infinitesimal, we have \lim_{\Delta x \to 0} \varepsilon = 0 (treating and as given so that is a function of alone).
Article summary
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function is differentiable at and that is infinitesimal. Then \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x for some infinitesimal , where \Delta y=f(x+\Delta x)-f(x). If \Delta x \neq 0 then we may write \frac{\Delta y}{\Delta x} = f'(x) + \varepsilon, which implies that \frac{\Delta y}{\Delta x}\approx f'(x) , or in other words that \frac{\Delta y}{\Delta x} is infinitely close to f'(x) , or f'(x) is the standard part of \frac{\Delta y}{\Delta x} . A similar theorem exists in standard Calculus. Again assume that is differentiable, but now let be a nonzero standard real number. Then the same equation \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x holds with the same definition of , but instead of being infinitesimal, we have \lim_{\Delta x \to 0} \varepsilon = 0 (treating and as given so that is a function of alone).
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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