Suppose f(x) is differentiable at x=a. What does tangent line approximation, y=, mean? Select all that apply
(A) Local linearization
(B) y=f(x)-f(a)-f'(a)(x-a)
(C) The best liner approximation of f(x) near a
(D) After zooming y is a good approxiamtion to f(x)
(E) f(x)~y=f(a) + f(a)(x-a)
(F) The tangent line equation to the curve at the point (x,y)
(G) The slope of the tangent line
For this answer I say A, B, F, and G.Please tell me if I am not including any more or if I am wrong in any way.Thanks.
Are you sure you don't mean
(E) f(x)~ y = f(a) + f'(a)(x-a) ??
That would be one of several acceptable answers. A, C, D and F are also OK
Equation B makes no sense. Are you sure you copied it correctly?
Your answer is correct.
The tangent line approximation, y=, refers to the concept of local linearization, where a straight line is used to approximate the behavior of a function near a particular point.
Option A is correct because local linearization refers to the process of approximating a function using a tangent line at a specific point.
Option B is correct because the equation y=f(x)-f(a)-f'(a)(x-a) represents the tangent line approximation at point (a, f(a)).
Option F is correct because the tangent line equation represents the equation of the line that is tangent to the curve of the function at the point (x,y).
Option G is correct because the slope of the tangent line is the rate of change of the function at the point of tangency.
Options C, D, and E are not correct because they do not accurately describe the concept of tangent line approximation.