Let the curve y = f (x) satisfies the equation `(dy)/(dx)=1-1/x^2` and passes through the point `(2,7/2)` then the value of f(1) is

Options

A. `3`
B. `2`
C. `7/2`
D. `1`

Answer

To solve the problem, we need to find the function \( f(x) \) given that its derivative \( \frac{dy}{dx} = 1 - \frac{1}{x^2} \) and that it passes through the point \( (2, \frac{7}{2}) \). We will also need to find the value of \( f(1) \).

### Step 1: Integrate the derivative
We start with the equation:
\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]
To find \( f(x) \), we need to integrate both sides with respect to \( x \):
\[ y = \int \left(1 - \frac{1}{x^2}\right) dx \]

### Step 2: Perform the integration
The integral can be computed as follows:
\[ y = \int 1 \, dx - \int \frac{1}{x^2} \, dx \]
\[ y = x + \frac{1}{x} + C \]
where \( C \) is the constant of integration.

### Step 3: Use the initial condition to find \( C \)
We know that the curve passes through the point \( (2, \frac{7}{2}) \). Thus, we substitute \( x = 2 \) and \( y = \frac{7}{2} \) into the equation:
\[ \frac{7}{2} = 2 + \frac{1}{2} + C \]
\[ \frac{7}{2} = 2 + 0.5 + C \]
\[ \frac{7}{2} = \frac{4}{2} + \frac{1}{2} + C \]
\[ \frac{7}{2} = \frac{5}{2} + C \]
Now, solving for \( C \):
\[ C = \frac{7}{2} - \frac{5}{2} = 1 \]

### Step 4: Write the function \( f(x) \)
Now we can write the function:
\[ f(x) = x + \frac{1}{x} + 1 \]

### Step 5: Find \( f(1) \)
Now we need to find \( f(1) \):
\[ f(1) = 1 + \frac{1}{1} + 1 = 1 + 1 + 1 = 3 \]

### Conclusion
Thus, the value of \( f(1) \) is:
\[ \boxed{3} \]