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.
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
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 byThis curve has a straightforward function that defines it.not to a function
Grandad
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