I have $f(x,y)=2x+y$ and i have trouble to find $u(t)$, the tangent vector for every level curve of $f$.
On the other side, i have the ellipse $4x^2+y^2=4$ and i must parameterize it in terms of $t$ (probably with $x=\cos(t), y=2\sin(t))$ to find $v(t)$ the tangent vector for every instant $t$.
This took me more time than I would like to admit, so a hint will be welcome.
Solved!
For the first problem: The level curves have the equation $f(x,y)=c$ where $c$ is a constant.
Then: $$f(x,y)=2x+y=c$$ $$y=c-2x$$
if i parametrize the level curves with $x=t$ i end with
$$r(t)=(t, k-2t)$$
and the tangent will be
$$u(t)=r'(t)=(1,-2)$$
For the ellipse $4x^2+y^2=4 \to x^2+ \frac{y^2}{4}=1$ i use the parametrization that i have mentioned before:$$g(t)=(\cos(t), 2\sin(t)) \ \ 0\le t \le 2\pi$$ and the tangent is $$v(t)=g′(t)=(-\sin(t),2 \cos(t))$$
If i did something wrong, please feel free to correct me.