Student Question

`x=t^2-t+2 , 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 when its slope `dy/dx` is zero, i.e. `dy/dt=0` and `dx/dt!=0` .

Curve has a vertical tangent if its slope approaches infinity i.e. `dx/dt=0` and `dy/dt!=0`

Given equations of the parametric curve are:





For horizontal tangents:






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

For t=1,



For t=-1,



Horizontal tangents are at the points (2,-2) and (4,2)

For vertical tangents,




Corresponding points on the curve for `t=1/2` are,









Vertical tangent is at the point `(7/4,-11/8)`

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