Answer:
Horizontal tangent
(x, y) = (1, 0)
Vertical tangent
(x, y) = DNE
Step-by-step explanation:
The equation for the slope (m) of the tangent line at any point of a parametric curve is:
[tex]m = \frac{\frac{dy}{dt} }{\frac{dx}{dt} }[/tex]
Where [tex]\frac{dx}{dt}[/tex] and [tex]\frac{dy}{dt}[/tex] are the first derivatives of the horizontal and vertical components of the parametric curves. Now, the first derivatives are now obtained:
[tex]\frac{dx}{dt} = -1[/tex] and [tex]\frac{dy}{dt} = 2\cdot t[/tex]
The equation of the slope is:
[tex]m = -2\cdot t[/tex]
As resulting expression is a linear function, there are no discontinuities and for that reason there are no vertical tangents. However, there is one horizontal tangent, which is:
[tex]-2\cdot t = 0[/tex]
[tex]t = 0[/tex]
The point associated with the horizontal tangent is:
[tex]x = 1 - 0[/tex]
[tex]x = 1[/tex]
[tex]y = 0^{2}[/tex]
[tex]y = 0[/tex]
The answer is:
Horizontal tangent
(x, y) = (1, 0)
Vertical tangent
(x, y) = DNE
