Calculate total cost (disregarding any fixed costs) or total...

Calculate total cost (disregarding any fixed costs) or total profit. Photos from Nature has found that its marginal cost, in cents per card, is $$C^{\prime}(x)=-0.04 x+85, \quad \text { for } x \leq 1000$$ where $x$ is the number of cards produced. Find the total cost of producing 650 cards.

Calculate total cost (disregarding any fixed costs) or total profit.
Photos from Nature has found that its marginal cost, in cents per card, is
$$C^{\prime}(x)=-0.04 x+85, \quad \text { for } x \leq 1000$$
where $x$ is the number of cards produced. Find the total cost of producing 650 cards.

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Key Concepts

Marginal Cost Function

The marginal cost function represents the rate of change of the total cost with respect to output. It shows the cost of producing one additional unit and is mathematically the derivative of the total cost function with respect to quantity.

Integration to Find Total Cost

Integration is used to accumulate the marginal costs over a range of production levels to compute the total cost. By integrating the marginal cost function from a starting production amount (often zero) to the desired production quantity, one obtains the sum of all variable costs incurred.

Definite Integration

Definite integration is the process of integrating a function over a specific interval. In cost calculations, it is used to determine the exact total cost associated with producing a given number of units by evaluating the integral with the appropriate lower and upper bounds.

Variable Cost Analysis

Variable cost analysis involves focusing on costs that change with the level of production, excluding any fixed costs that remain constant regardless of output. In this context, only the expenses that vary with production (reflected by the marginal cost) are considered.

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