Your selection of A, B, F, and G is correct.
(A) Local linearization refers to the process of approximating a function around a point using a tangent line. This approximation is valid for small intervals near the given point.
(B) The equation y = f(x) - f(a) - f'(a)(x - a) represents the tangent line approximation. It calculates the difference between the function f(x) and its linear approximation at the point (a, f(a)).
(F) The tangent line equation to the curve at the point (x, y) is another way to express the tangent line approximation. It represents the line that touches the curve at the specific point (x, y).
(G) The slope of the tangent line refers to the derivative of the function at the point of tangency. It represents the rate of change of the function at that point.
(C), (D), and (E) are not equivalent to the tangent line approximation.
(C) The best linear approximation of f(x) near a refers to the linear function that best approximates the original function in a neighborhood around the point a.
(D) After zooming, y being a good approximation to f(x) implies that as you zoom in on the graph around the point (a, f(a)), the tangent line becomes a better approximation to the function.
(E) The equation f(x)~y = f(a) + f'(a)(x - a) represents the linearization of the function, which is a good approximation near the point a based on the tangent line.
So, the correct options are A, B, F, and G.