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In economics, understanding the total cost function is crucial for businesses to evaluate their production expenses. A total cost function represents the sum of all costs incurred in producing a certain quantity of goods. This includes both fixed costs and variable costs.
To find the total cost for producing mountain bikes in this context, we start with the marginal cost function, which shows the cost of producing one additional unit. It is given by \(C'(q) = \frac{600}{0.3q+5}\). By integrating this marginal cost function with respect to \(q\), we derive the total cost function \(C(q)\).
The integration of \(C'(q)\):

• Leads to \(C(q) = 2000 \ln(0.3q + 5) + C\), where \(C\) is an integration constant.
• The fixed cost, represented by \(C(0) = 2000\), helps determine this constant.
• The complete cost function becomes \(C(q) = 2000 \ln(0.3q + 5) + 2000 - 2000 \ln(5)\).

This function helps calculate the total costs for producing any given number of bicycles.

Fixed costs are those expenses that do not vary with the level of production. They are incurred even if the company produces nothing. For bike production, the fixed cost is given as $2000.
Fixed costs include:

• Rent for factory or warehouse space
• Salaries of permanent staff
• Equipment depreciation

These costs remain constant regardless of how many bicycles are produced.
In our exercise, the fixed cost is directly included when calculating the total cost function. By integrating the marginal cost function and using the known fixed cost, we set the constant in our total cost equation. This integration ensures that the fixed costs are always part of the total cost, maintaining consistency across different levels of production.

Marginal profit is key to understanding how much profit is earned from producing one additional unit of a good. It is the difference between marginal revenue and marginal cost. In this scenario, the marginal revenue for each bike is the selling price, which is \($200\).
Marginal profit calculation steps involve:

• Calculating the marginal cost using the given function \(C'(q)\).
• For the 31st bike, the marginal cost is \(\frac{600}{0.3 \times 31 + 5} \approx 41.96\).
• The marginal profit is then the difference: \(200 - 41.96 = 158.04\).

This figure represents the additional profit made by selling the 31st bicycle, demonstrating the financial benefit of increasing production by one unit.
This concept helps businesses make informed decisions about whether to continue increasing their production levels.

Integration in calculus is a fundamental process used to find functions that describe accumulated quantities. It calculates the total accumulation of many small factors, which is useful in determining concisely the total cost from marginal cost.
The integration process:

• Starts with a function like \(C'(q)\), representing the rate of change as a function of \(q\) (quantity produced).
• Involves finding the antiderivative, which includes a constant of integration \(C\), denoting unknown fixed costs or initial conditions.
• In our bike example, integrating \(\frac{600}{0.3q+5}\) results in finding \(C(q)\), the total cost function.

This method enables the transformation of a dynamic rate (marginal cost) into a complete snapshot (total cost function) that provides insight into cumulative costs over a range of production quantities.
Integration is thus essential for converting individual rate-based functions into comprehensive cost evaluations necessary for business calculations.