Student Question

`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:

`x=t+4`

`y=t^3-3t`

`dx/dt=1`

`dy/dt=3t^2-3`

For Horizontal tangents,

`dy/dt=0`

`3t^2-3=0`

`=>3t^2=3`

`=>t^2=1`

`=>t=+-1`

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

For t=1,

`x_1=1+4=5`

`y_1=1^3-3(1)=-2`

For t=-1,

`x_2=-1+4=3`

`y_2=(-1)^3-3(-1)=2`

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

For vertical tangents,

`dx/dt=0`

However `dx/dt=1!=0`

So the curve has no vertical tangents.

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