The problem bellow is a subquestion from an exercize of a previous exam in university of Athens.
Let $R(u,v)=(u,v,u^2+v^2)$ be a paraboloid. Show that $w=(1,1,2)$ is tangent to $R(q)$ ,$q=(1,0)$.Also find the curve $r(t)=R(u(t),v(t))$ such as $r(0)=R(q)=(1,0,1)$ and $r'(0)=w$\
What I have done so far :
$W$ is in a tangent vector at $q$ because it belongs to $q$'s tangent space $T_qR$ ,since we can observe that $w=R_u(q)+R_v(q)$.
For the rest of the question. I've tried solving the problem in the form of of a differential equation but of as expected it lead nowhere since I dont have enough data to proceed or that's what I think at least.That's where im stuck right now : $r(t)=(u(t),v(t),u^2(t)+v^2(t))$, so $u(0)=1 , v(0)=0 and u^2(0)+v^2(0)=1$
and hence $u'(0)=1,v'(0)=1 $
My guess is that's a wrong way to proceed but it was the only thing that came to mind .Any help or tip would be really appreciated ,thank you in advance !