Final answer:
The slope of the line tangent to the function f(x) = 3sin(x) / (3x+3) at x=0 is m = 1/2.
Step-by-step explanation:
To find the slope of the tangent line to the function f(x) at x=0, we need to find the derivative of f(x) with respect to x. The derivative of f(x) can be calculated using the quotient rule, which states that if f(x) = u(x) / v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / [v(x)]^2.
Given f(x) = 3sin(x) / (3x+3), we'll use the quotient rule where u(x) = 3sin(x) and v(x) = 3x+3. The derivatives u'(x) and v'(x) are u'(x) = 3cos(x) and v'(x) = 3, respectively. Plugging these values into the quotient rule formula:
f'(x) = [(3cos(x))(3x+3) - (3sin(x))(3)] / [(3x+3)^2]
To find the slope of the tangent line at x=0, we'll evaluate f'(x) at x=0:
f'(0) = [(3cos(0))(3(0)+3) - (3sin(0))(3)] / [(3(0)+3)^2]
f'(0) = [(3)(3) - (0)(3)] / [3^2]
f'(0) = 9 / 9
f'(0) = 1
Therefore, the derivative of the function f(x) evaluated at x=0 gives us the slope of the tangent line at that point, which is m = 1/2.