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 rate of change of the total cost with respect to the level of production or output. It is the derivative of the cost function and shows how costs will change when production levels are adjusted.

Integration (Finding the Cost Function)

To obtain the total cost function from the marginal cost function, one integrates the marginal cost function with respect to the level of production. This process finds the antiderivative, which gives a general form of the cost function that includes an arbitrary constant.

Fixed Cost

Fixed costs are the costs that do not change with the level of production and are incurred even when output is zero. They are used to determine the integration constant when integrating the marginal cost function, by setting the cost function equal to the fixed cost at zero output.

Transcript

00:01 So given that we want to find the cost function based off the marginal cost with the marginal cost function, c prime of x is equal to 0 .2x squared plus 5x, and we're told that our fixed cost is $10.

00:22 So we might need to recall a couple of things.

00:26 So it's actually written right here that we're given c prime of x, which is our marginal.

00:34 But just remember that marginal means a very small change, or in terms of what calculus looks at, is the derivative of the cost function.

00:46 So even if they don't tell you that this here is the derivative of cost, we should be able to know that this function here is actually just the derivative of the cost function.

00:58 And also what it means by fixed cost is if i plug in zero into our cost function c once we find it, it should spit out $10.

01:15 So if i want to first find one possible c, we would want to take the anti -derivative of this function with respect to x.

01:24 So let me go ahead and rewrite this.

01:27 So i get c prime of x is equal to 0 .2x squared plus 5x.

01:37 So i want to integrate each side of this function with respect to x first.

01:48 So the integral of c prime of x will just be c of x.

01:56 And then to find this integral here, we'll want to go ahead and we'll notice that both of these can be solved using the power rule.

02:09 So i can go ahead and use the sum and scalar property of the integral to first rewrite this like so.

02:17 So i'll go ahead and write a little space in between everything.

02:22 So 5x.

02:24 And then i go ahead and put in my integration symbols.

02:27 Oops, wrong color.

02:30 So now i'll go ahead and put in my colors not wanting to work well with me.

02:34 Right, there we go.

02:36 So put in my integration symbol around my variables, and now i get to apply the power rule twice.

02:43 So i have 0 .2x squared, and then the power rule says add one to the power, and then divide by the new power, 3, and then i need to add on a constant of integration...

Please give Ace some feedback