After finishing up the section called Tangent Planes and Linear Approximations, I thought I knew the material, until I began working on the first problem.
This portion of the exercise reads as follows:
For the following exercises, as a useful review of techniques used in this section, find a normal vector and a tangent vector at the point $P$.
This section never covered tangent vectors. Not only that, but this book has not touched finding vectors using implicitly defined functions. Here is the implicitly defined curve
$$x^2 + xy + y^2 - 3 = 0$$
and we want to find the normal and tangent vectors at the point $(-1,-1)$.
I don't understand what's going on or how to approach this problem. This section is before the concept of a gradient, which I heard would help in solving this problem, but I want to try and understand what to do with the given information independent from the gradient.
Question I would like to know the intuition behind solving this problem, because I think if I had that, I would at least know want I'm supposed to be building instead of taking derivatives aimlessly. In fact, using implicit differentiation and evaluating the derivative at $(-1,-1)$, I get
$$\frac{dy}{dx} = -1$$
But I don't know what to do with it.
If you pretend you could draw the graph, it would maybe look like this, right? (Ignore the coordinate axes)
Maybe i drew the curve wrong, but I got the tangent line correct, right?
What I'm trying to say is, as long as $dy/dx$ makes sense, it is indeed the gradient of the tangent line to the curve at that point.
Once I know a tangent vector, I can find a normal vector by rotation.