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What is the true meaning of a tangent in mathematics?
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[QUOTE="Greg Bernhardt, post: 6483600, member: 1"] From [USER=572553]@fresh_42[/USER]'s Insight [URL]https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/[/URL] Please discuss! Yes, it is the derivative of ##y.## But what is meant by that? Obviously we have a function ##x \longmapsto y=y(x)## and a derivative $$y'=y'(x)=\dfrac{dy}{dx}=\left. \dfrac{d}{dx}\right|_{x=a}y(x)=y(a+h)-J(h)-r(h)=y'(a) $$ It now isn't obvious at all what is meant: the function ##x\longmapsto y'(x)##, the value of the slope ##y'(a)##, or the linear map ##J,## the Jacobi matrix, the tangent in a way? Fact is, all of them, as needed according to the situation. I don't say we should teach tangent bundles and sections, but a little bit more accuracy would smoothen the step to calculus at college. [/QUOTE]
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What is the true meaning of a tangent in mathematics?
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