If (-1,2) and and (2,4) are two points on the curve y=f(x) and if g(x) is the gradient of the curve at point (x,y) then the value of the integral `int_(-1)^(2) g(x) dx` is

Options

A. 2
B. -2
C. 0
D. 1

Answer

To solve the given problem, we need to evaluate the integral of the gradient function \( g(x) \) over the interval from -1 to 2. Here are the steps to arrive at the solution:

### Step 1: Understand the given points
We are given two points on the curve \( y = f(x) \):
- Point 1: \((-1, 2)\) implies \( f(-1) = 2 \)
- Point 2: \((2, 4)\) implies \( f(2) = 4 \)

### Step 2: Define the gradient function
The gradient function \( g(x) \) is defined as the derivative of \( f(x) \):
\[ g(x) = f'(x) \]

### Step 3: Set up the integral
We need to evaluate the integral:
\[ \int_{-1}^{2} g(x) \, dx \]
Since \( g(x) = f'(x) \), we can rewrite the integral as:
\[ \int_{-1}^{2} f'(x) \, dx \]

### Step 4: Apply the Fundamental Theorem of Calculus
According to the Fundamental Theorem of Calculus, the integral of a derivative over an interval gives us the difference of the function evaluated at the endpoints of the interval:
\[ \int_{a}^{b} f'(x) \, dx = f(b) - f(a) \]
In our case, \( a = -1 \) and \( b = 2 \):
\[ \int_{-1}^{2} f'(x) \, dx = f(2) - f(-1) \]

### Step 5: Substitute the values
From the points given, we know:
- \( f(2) = 4 \)
- \( f(-1) = 2 \)

Now substituting these values into the equation:
\[ \int_{-1}^{2} f'(x) \, dx = 4 - 2 \]

### Step 6: Calculate the result
\[ 4 - 2 = 2 \]

Thus, the value of the integral \( \int_{-1}^{2} g(x) \, dx \) is:
\[ \boxed{2} \]