What is the true meaning of a tangent in mathematics?

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In summary, the conversation discusses the concept of derivatives and how it is commonly misunderstood in mathematics education. The term "derivative" can refer to the function itself, the value of the slope at a specific point, or the linear map of the tangent line. The lack of precision in understanding this concept can lead to confusion and mistakes in calculus problems.
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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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In the USA, a calculus problem asking for the tangent of ##f(x)## at ##x = a## is understood to ask for the equation of a line tangent to the graph at ##f(x)## at ##x = a## so ##f'(x)|_{x=a}## is a wrong answer.
 

FAQ: What is the true meaning of a tangent in mathematics?

1. What is a tangent?

A tangent is a mathematical concept that represents a straight line that touches a curve at a single point, without intersecting it. It is also used to describe the ratio of the opposite side to the adjacent side in a right triangle.

2. How is a tangent different from a secant?

A tangent and a secant are both lines that intersect a curve, but a tangent only touches the curve at one point while a secant intersects the curve at two points.

3. What is the relationship between a tangent and a circle?

In geometry, a tangent line is always perpendicular to the radius of a circle at the point of tangency. This means that the tangent line touches the circle at only one point and is at a 90-degree angle to the radius at that point.

4. Can a tangent line be parallel to a curve?

No, a tangent line cannot be parallel to a curve. By definition, a tangent line touches the curve at one point, so it cannot be parallel to it.

5. How is the concept of a tangent used in real life?

The concept of a tangent is used in various fields such as engineering, physics, and astronomy. For example, in engineering, tangents are used to determine the slope of a curved road or a bridge. In physics, tangents are used to calculate the rate of change of a moving object's position. In astronomy, tangents are used to calculate the angle of elevation of a celestial object.

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