Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents x=t^3-3ty=3t^2-9

Nann 2021-09-12 Answered
Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents \(\displaystyle{x}={t}^{{3}}-{3}{t}\)
\(\displaystyle{y}={3}{t}^{{2}}-{9}\)

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Expert Answer

Szeteib
Answered 2021-09-13 Author has 5277 answers
\(\displaystyle{x}={t}^{{3}}-{3}{t}\)
\(\displaystyle{y}={3}{t}^{{2}}-{9}\)
\(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{6}{t}}}{{{3}{t}^{{2}}-{3}}}\)
for horizontal tangent \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={0}\Rightarrow{t}={0},{P}{\left({0},-{9}\right)}\)
For verticle tangent \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\infty{o}{t}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{1}}{{0}}\)
\(\displaystyle\frac{{{6}{t}}}{{{3}{t}^{{2}}-{3}}}=\frac{{1}}{{0}}\Rightarrow{t}=\pm{1}\)
\(\displaystyle{P}{\left(-{2},-{6}\right)}\phi{\left({2},-{6}\right)}\)
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