Horizontal Tangent in Parametric Equations: Finding & Understanding

[mill]

If the tangent is horizontal, it is where the tangent is zero. In single var. calc. that would be at max. or min. for example. I am confused about what horizontal tangent refers to when I am given a parametric equation.

E.g. At what value of t does x=t^2 -t and y=t^2 +t have a horizontal tangent?

The answer is -1/2 which can be found by setting y'=0. I don't understand why this happens though. As in, why dy/dt rather than dy/dx or why does dx/dt not apply? In describing the curve, what is the relationship between the two (x and y given in parametric form) that I could just only look at dy/dt?

My first instinct was to look for dy/dx which would look something like (2t+1)/(2t-1).

 

[Adithyan]

The answer to your question is pretty simple.

\frac{dy}{dx} =\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.

So for \frac{dy}{dx} to be zero, the numerator i.e \frac{dy}{dt} must be zero. And hence the answer.

 

[mill]

Thanks.

 

[SteamKing]

If the tangent is horizontal, it is where the tangent is zero.

More accurately, if the tangent is horizontal, it is where the slope of the tangent line is zero.