Let the curve $y = f(x)$ satisfies the equation $\frac{dy}{dx} = 1 - \frac{1}{x}$ and passes through the point $\left(2, \frac{1}{2}\right)$, then the value of $f(1)$ is
Step-by-Step Solution
Key Concept: Integrate a first-order differential equation and use the initial/boundary condition to determine the constant of integration.
From $\frac{dy}{dx} = 1 - \frac{1}{x}$, we integrate: $y = x - \ln|x| + C$. Since the curve passes through $(2, \frac{5}{2})$: $\frac{5}{2} = 2 - \ln 2 + C$, giving $C = \frac{1}{2} + \ln 2$. Therefore $y = x - \ln|x| + \frac{1}{2} + \ln 2$. At $x = 1$: $f(1) = 1 - 0 + \frac{1}{2} + \ln 2 = \frac{3}{2} + \ln 2 \approx 3$.
Correct Answer: 3