I What is the true meaning of a tangent in mathematics?

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The discussion centers on the concept of a tangent in mathematics, specifically its relationship to derivatives. It highlights that the term "tangent" can refer to various mathematical constructs, including the function mapping, the slope at a specific point, and the Jacobi matrix. The conversation suggests that a more precise understanding of these concepts would enhance the transition to calculus in educational settings. In the U.S., students often misinterpret calculus problems regarding tangents, mistakenly providing only the derivative rather than the equation of the tangent line. A clearer explanation of tangents could improve comprehension and application in calculus.
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From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

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.
 
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