Business and Economics 45 - 52 cost Find the cost function for each marginal cost function C^'(x

Business and Economics 45 - 52 cost Find the cost function...

Key Concepts

Marginal Cost Function

The marginal cost function represents the instantaneous rate of change of the total cost with respect to the number of units produced. It is essentially the derivative of the cost function and provides insight into how costs are affected by producing one additional unit.

Integration (Antidifferentiation)

Integration is the mathematical process of finding the antiderivative of a function. When provided with a marginal cost function, integrating it with respect to the quantity yields the total cost function up to an arbitrary constant, which can then be determined using additional information.

Cost Function

The cost function represents the total cost of producing x units of a product. It is obtained by integrating the marginal cost function and accounts for all costs incurred during production, providing a cumulative measure of expenses.

Initial Condition Determination

Initial conditions, such as a specified cost for a given number of units, are used to determine the constant of integration in the cost function. By substituting the known cost and production level into the integrated function, the arbitrary constant can be solved, ensuring the cost function accurately reflects the actual costs.

Transcript

00:01 So given the fact that we have, or if we want to find the cost function, given that the marginal cost is, so c prime of x is equal to x plus 1 over x squared, and that two units will cost $5 .50 to make.

00:25 Well, so even if we weren't told that our marginal cost was the derivative of the cost function, where we could.

00:34 Do is recall that what marginal means is a very small change of something.

00:43 So this is saying a very small change in the cost function and the way we look at very small changes in calculus is by the by the derivative.

01:07 So and the unit cost being or to produce two units costing by 50 will be, so this means if our cost function is c, and c of two should be $5 .56.

01:34 So if i want to find one possible c or one possible cost function, i can go ahead and integrate this first, or integrate our marginal cost.

01:45 So let's go ahead and write that now.

01:46 So i have c prime of x is equal to x plus.

01:53 So i can rewrite 1 over x squared as x to the negative second power, just because we are going to end up taking the integral of this, or antiderivative.

02:06 So i'll just go ahead and rewrite it like this first.

02:09 So when we do take that anti -derivative, it will be a little bit easier for us to do so.

02:14 So now i'm going to integrate each size.

02:18 Of this, so i'll need to move my equal sign over a little bit so i can squeeze in a dx there.

02:25 So i integrate each side with respect to x and now the anti -derivative of the derivative or the marginal cost will just be our cost function...

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