Questions: The marginal cost function C'(x) is shown below, where x is in thousands of units. If the total cost of producing 2 thousand units is C(2)=3100, use the Net Change concept to determine the total cost of producing 3 thousand units.

Solution Steps

The problem provides the marginal cost function \( C'(x) \) and asks us to determine the total cost of producing 3 thousand units given that the total cost of producing 2 thousand units is $3100. We will use the Net Change concept to solve this.

From the graph, the marginal cost function \( C'(x) \) appears to be a linear function. We need to determine the equation of this line. The line starts at \( (0, 20) \) and goes through \( (5, 0) \).

Using the two points \( (0, 20) \) and \( (5, 0) \), we can find the slope \( m \) of the line: \[ m = \frac{0 - 20}{5 - 0} = -4 \]

The equation of the line in slope-intercept form is: \[ C'(x) = -4x + 20 \]

To find the total cost function \( C(x) \), we integrate the marginal cost function \( C'(x) \): \[ C(x) = \int (-4x + 20) \, dx = -2x^2 + 20x + C \]

We know that \( C(2) = 3100 \). Plugging in \( x = 2 \): \[ 3100 = -2(2)^2 + 20(2) + C \] \[ 3100 = -8 + 40 + C \] \[ 3100 = 32 + C \] \[ C = 3100 - 32 = 3068 \]

So, the total cost function is: \[ C(x) = -2x^2 + 20x + 3068 \]

Now, we need to find \( C(3) \): \[ C(3) = -2(3)^2 + 20(3) + 3068 \] \[ C(3) = -18 + 60 + 3068 \] \[ C(3) = 3110 \]

Final Answer