[Math] the tangent at a sharp point on a curve

tangent line

How to know which line represents tangent to a curve $y=f(x)$ (in RED) ?From the diagram , I cannot decide which line to take as tangent , all seem to touch at a single point.

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None of them, because it's not differentiable there, so there is no tangent line at that point.

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[Math] Proof of tangent lines to a curve

Quick answer: the tangent line may intersect the curve many times. For general curves, the tangent line at a point cannot be defined reasonably by counting the intersection points. Just think of the graph of $x \mapsto \sin x$ at $x=\pi/2$: the tangent line is the line $y=1$, which touches the graph infinitely many times.

What you say can be made rigorous for very particular curves, like conic sections.

On the other side, it all boils down to your definition of tangent line. There are situations where mathematicians prefer to allow a whole bunch of tangent lines, although there is no single straight line that can be pointed out as the tangent line. In elementary treatments, however, differentiability is somehow equivalent to the existence of a tangent line.

[Math] For a curve, find the unit tangent vector and parametric equation of the line tangent to the curve at the given point

Assuming you are right about knowing how to find $T(t)$( I believe in you, you can do it!) Recall that a line can be parameterized as $l(t)=v_0+tv$ where $v_0$ is a position point(i.e. the value of your function at the point you want to find the tangent) and $v$ the direction vector(i.e. the "slope" of your line, so the tangent to your curve at the desired point.)

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Related Solutions

[Math] Proof of tangent lines to a curve

[Math] For a curve, find the unit tangent vector and parametric equation of the line tangent to the curve at the given point

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