To solve for f(3) and f'(3), let’s start by understanding the situation with a tangent line. We know that the tangent line to the function y = f(x) at the point (3, 2) indicates that:
1. The value of the function at x = 3 is f(3) = 2.
2. The slope of the tangent line, denoted as f'(3), is essential in determining how the function behaves around the point (3, 2).
Since the tangent line passes through the points (3, 2) and (0, 1), we can calculate the slope of that line. The slope (m) between the two points (x1, y1) = (3, 2) and (x2, y2) = (0, 1) is given by the formula:
m = (y2 – y1) / (x2 – x1)
Substituting the coordinates, we get:
m = (1 – 2) / (0 – 3) = -1 / -3 = 1/3
This slope corresponds to the derivative of the function at that point: f'(3) = 1/3.
In summary, we have found the values:
- f(3) = 2
- f'(3) = 1/3
Therefore, the function is defined at the point and the slope of the tangent line is adequately described at the point of tangency. This information can further help us understand the behavior of the function around this point.