Study of a parametric curve

Question

geniuses.

We want to study this parametric curve in the neighborhood of zero : $$ \left\{ \begin{array}{r c l} x(t) &=& t - t^3 + 2t^4\\ y(t) &=& -2t + 2t^3 + t^5 \\ \end{array} \right. \ $$

We want to extract as much information as possible from this parameters. For example, what kind of point is it? Inflection point ? And the curve relative position to the tangent.

We want to extract these information using the Taylor Expansion/Serie formulas.

Thank you for your help and explanation, stackExchange explanations are really powerful to understand mathematics a bit better.

Answer 1

Well, Taylor expansion is already done :-) You can write it $$\begin{pmatrix}x(t) \\ y(t)\end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix} + \begin{pmatrix}1 \\ -2\end{pmatrix}t + \begin{pmatrix}0 \\ 0\end{pmatrix}t^2 + \begin{pmatrix}-1 \\ 2\end{pmatrix}t^3 + \begin{pmatrix}2 \\ 0\end{pmatrix}t^4 + \begin{pmatrix}0 \\ 1\end{pmatrix}t^5 + o(t^5)$$ so

• The curve passes threw the point $(0,0)$
• It is tangent to vector $(1,-2)$
• Vector $(-1,2)$ is collinear to $(1,-2)$, so nothing new
• the first non-colinear vector is $(2,0)$, so the curve presents a simple stationary point at $(0,0)$ (first direction is odd, second is even)
• don't know what to extract from the $t^5$ vector...