Suppose that the graph of `y=f(x)`, contains the points `(0,4) and (2,7)`. If `f'` is continuous then `int_(0)^(2) f'(x) dx` is equal to

Options

A. `2`
B. `-2`
C. `3`
D. none of these

Answer

To solve the problem, we need to evaluate the definite integral of the derivative of the function \( f(x) \) over the interval from 0 to 2. Given that the graph of \( y = f(x) \) passes through the points (0, 4) and (2, 7), we can use this information to find the value of the integral.

### Step-by-Step Solution:

1. **Identify the Points:**
We have two points on the graph of \( f(x) \):
- \( (0, 4) \) which means \( f(0) = 4 \)
- \( (2, 7) \) which means \( f(2) = 7 \)

2. **Calculate the Slope:**
The slope of the line connecting these two points can be calculated using the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) = (0, 4) \) and \( (x_2, y_2) = (2, 7) \):
\[ \text{slope} = \frac{7 - 4}{2 - 0} = \frac{3}{2} \]
This slope represents the average rate of change of the function \( f(x) \) over the interval [0, 2], which is equal to \( f'(x) \) since \( f' \) is continuous.

3. **Set Up the Integral:**
We need to evaluate the integral:
\[ \int_{0}^{2} f'(x) \, dx \]
By the Fundamental Theorem of Calculus, this integral gives us the net change in \( f(x) \) from \( x = 0 \) to \( x = 2 \):
\[ \int_{0}^{2} f'(x) \, dx = f(2) - f(0) \]

4. **Substitute the Values:**
Now we can substitute the known values of \( f(2) \) and \( f(0) \):
\[ f(2) - f(0) = 7 - 4 = 3 \]

5. **Final Result:**
Therefore, the value of the integral is:
\[ \int_{0}^{2} f'(x) \, dx = 3 \]

### Conclusion:
The final answer is \( 3 \).