Can Linearization from Calculus I be used as a Multivariable function? In my math class yesterday we learned about linearization of a function to approximate values in a small range of x values (followed by Newton's method of approximating the zeros of functions). We were given the general formula,
$$L(x) = f(a)+f^\prime(a)(x-a)$$
Where a (usually an integer) a value 'close enough' to represent the actual value of x you want know
I ran over this concept in my head through the rest of the day, and I came to a question. What if one considered all values of a and x for a given function. Essentially,
$$L(x,a) = f(a)+f^\prime(a)(x-a)$$
Ex. Let $f(x) = x^3 + 2x+1$ (normally the function is more complex but I needed something with fairly simplistic algebra)
We would then have
$$f^\prime(x) = 3x^2+2$$
Suppose $a$ represents all possible approximation values,
$$L(x, a)=a^3+2a+1+(3a^2+2)(x-a)$$
Simplifying,
$$= a^3+2a+1+3a^2x+2x-3a^3-2a$$
$$\downarrow$$
$$= -2a^2+3a^2x+2x+1$$
Is this kind of function useful?

It's potentially useful in that it gives you all the linear approximations at once,  but no, I don't think you'd often want to deal with it in practice. I don't know of a context where your $L(x,a)$ would be useful (but I'd be happy to be proven wrong by another answer).
The linear approximation is typically bad when $x$ is far from $a$ - you don't actually care what the tangent line at $a=5$ thinks might be the value of the function at $x=173.2$. So in practice, you'd likely be taking linear approximation at a well-chosen $a$ near your value of interest, just as you describe.
When we are interested in generic information rather than linear approximation at a point, then note that this basically just combines the information of $f(a)$ and $f'(a)$ in a standard way, so that we may as well keep the flexibility of the two separate functions/values.
