Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/3x-1; a= 2

Verified step by step guidance

1. Step 1: Identify the function f(x) = \frac{1}{3}x - 1 and the point a = 2 where you need to find the derivative f'(a).
2. Step 2: Recall that the derivative f'(x) of a linear function f(x) = mx + b is simply the coefficient m of x. In this case, f(x) = \frac{1}{3}x - 1, so the derivative f'(x) is \frac{1}{3}.
3. Step 3: Since the derivative of a linear function is constant, f'(x) = \frac{1}{3} for all x. Therefore, f'(a) = f'(2) = \frac{1}{3}.
4. Step 4: Interpret the result: The derivative f'(a) = \frac{1}{3} represents the slope of the tangent line to the graph of f(x) at the point where x = 2.
5. Step 5: Conclude that the slope of the tangent line at x = 2 is \frac{1}{3}, which means the line rises \frac{1}{3} unit for every 1 unit it moves to the right.