what's exactly a rigorous definition of

tangent & normal to a curve suitable in all cases?

I know its a dumb qusetion but I have problems

in using it for concepts of calculus.

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- Oct 21st 2008, 06:12 AMADARSHtangent & normal
what's exactly a rigorous definition of

tangent & normal to a curve suitable in all cases?

I know its a dumb qusetion but I have problems

in using it for concepts of calculus. - Oct 21st 2008, 07:09 AMJhevon
the tangent line to a curve at a particular point gives the slope (rate of change) of the curve at that point. it is a straight line that touches the curve once in that vicinity. the normal line at the said point is the straight line that is perpendicular to the tangent line, and so cuts through the curve

- Oct 22nd 2008, 03:52 AMADARSH...another one
Thank you

A line can both be tangent & normal to a curve

but not to a function

isn't it true. (Angel) - Jan 17th 2009, 04:23 PMJhevon
- Jan 17th 2009, 10:39 PMGrandadTangent and Normal
Hello ADARSH

In fact, there's nothing to stop a line being both a tangent and a normal to the same curve: it could be the tangent at one point, and normal to the curve at another point.

But perhaps that's not what you meant. So see the attachment, which shows the graph of the curve whose equation in polar coordinates is $\displaystyle r =\sin 2 \theta$. You'll see that each axis is both a tangent and a normal to the curve at the origin.

I'm not quite sure what you mean byQuote:

not to a function

Grandad - Jan 17th 2009, 10:48 PMADARSH
Well a function has only one value of y defined for any "x" but a curve as in your attatchment can have many "y" for a single "x" (Nod)

So I wanted to ask ,will that happen in both the cases which I don't think it will(Shake)

-Adarsh - Jan 17th 2009, 11:07 PMGrandadTangent and Normal
Hello ADARSHI see what you mean. I think you are right. If, for a given x, there is at most one value of y, there will be only (at most) one gradient of the curve at any given point. Hence a single line cannot be both a tangent and a normal to such a curve at the same point.

Grandad

- Jan 18th 2009, 03:00 AMADARSHyep
Yes that's what I meant,thank you for that(Happy)