Final answer:
To find the linearization of f(x) = sin(x) at a = 3, calculate f(3) and f'(3), then use the formula l(x) = f(a) + f'(a)(x - a) to get l(x) = sin(3) + cos(3)(x - 3).
Step-by-step explanation:
The student is asking for the linearization of the function f(x) = sin(x) at a = 3. The linearization l(x) is the linear approximation of the function around the point a. To find it, we need the function's value and its derivative at a. The derivative of sin(x) is cos(x), so we need to find f(3) and f'(3).
Therefore, f(3) = sin(3) and f'(3) = cos(3). The linearization of f at a=3 is given by:
l(x) = f(a) + f'(a)(x - a)
Plugging in the values, we get:
l(x) = sin(3) + cos(3)(x - 3)
This equation represents the tangent line at x = 3, which is the best linear approximation to the function f(x) = sin(x) around this point.