Let's solve the given problem step-by-step.

Problem Details

We are given:

• Cost Function: $C(x) = 50 + 0.01x^3$, which models the total cost of producing $x$ units of a product.
• Price Function: $p(x) = 100 - 0.1x$, which is the price function for $x$ units.

Part (a): Find the Marginal Cost Function

The marginal cost is the derivative of the cost function $C(x)$ with respect to $x$. It represents the cost of producing one additional unit.

$\text{Marginal Cost } C'(x) = \frac{d}{dx}(50 + 0.01x^3)$

Calculating the derivative:

$C'(x) = 0 + 0.01 \cdot 3x^2 = 0.03x^2$

So, the marginal cost function is:

$\boxed{C'(x) = 0.03x^2}$

Part (b): Set Up an Integral for the Total Revenue Function and Compute the Producer Surplus When 50 Units Are Produced

The total revenue function is given by multiplying the price function by the number of units produced:

$R(x) = p(x) \cdot x = (100 - 0.1x)x = 100x - 0.1x^2$

To find the producer surplus, we need to integrate the revenue function up to the production level and subtract the total cost.

Producer Surplus is given by:

$\text{Producer Surplus} = \int_{0}^{x} p(q) \, dq - C(x)$

For $x = 50$, the integral becomes:

$\text{Producer Surplus} = \int_{0}^{50} (100 - 0.1q) \, dq - C(50)$

1. Compute the Integral:

$\int (100 - 0.1q) \, dq = 100q - 0.05q^2$

Evaluating from 0 to 50:

$\left[100q - 0.05q^2 \right]_{0}^{50} = (100 \times 50 - 0.05 \times 50^2) - (100 \times 0 - 0.05 \times 0^2)$