How would I go about determining the point where two functions not intersect, but where they are tangent?
The functions (or rather curves) that I'd like to find that point for are $u^2 + v^2 = c$ and $v = \frac{b}{a}u + \frac{ag}{2} \frac{1}{u}$, where $c>0,a>0,b>0,g>0$.
My idea is the calculate the tangent line function $v(u) = mu + b$ for each of these and set them equal to each other. That is, find the conditions such $v_1(u_{tan}) = v_2(u_{tan})$. But if this is the case, why can't I just set the derivatives of these two functions to each other, i.e. $v_1' = v_2'$ and solve? I saw somewhere that this isn't the right way to do it, and it makes sense. In looking for these conditions, we are looking for the appropriate constants such that these functions are tangent to one other. Which is the way to go?
Both approaches fail to account for vertical tangents. Leaving that aside, examining the derivatives gives you the slopes of the tangents, but not their locations. You’ll find values of $u$ for which the tangents to the curves are parallel, but you still need some other constraint to ensure that the two lines coincide, namely, that $v_1(u)=v_2(u)$ as well.