`x=t+4 , y=t^3-3t` Find all points (if any) of horizontal and vertical tangency to the curve.

Expert Answers

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Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero, i.e when `dy/dt=0` and `dx/dt!=0`

Curve has a vertical tangent line, if its slope approaches infinity i.e `dx/dt=0`

and `dy/dt!=0`

Given parametric equations are:





For Horizontal tangents,






Corresponding points on the curve can be found by plugging in the values of t in the equations,

For t=1,



For t=-1,



Horizontal tangents are at the points (5,-2) and (3,2)

For vertical tangents,


However `dx/dt=1!=0`

So the curve has no vertical tangents.



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