The cost to produce c(q) = q^2 (√q - 2) items is dollars. Find the marginal cost.

Question

Just to be clear, this is what the question actually looks like:

Answer 1

Hey Lilas,

Understanding Marginal Cost

Marginal cost is a fundamental concept in economics and business. It represents the cost of producing one additional unit of a good. In mathematical terms, it's the derivative of the total cost function with respect to the quantity of goods produced. This derivative gives us the rate of change of the total cost with respect to a small change in quantity.

The Cost Function:

Our cost function is C(q)=q^2(sqrt(q)-2), where q is the quantity of items produced. This function tells us the total cost of producing q items.

Calculating Marginal Cost

To find the marginal cost, we need to differentiate C(q) with respect to q.

The Cost Function: C(q)=q^2(sqrt(q)-2)

We have our cost function C(q), and we can break it down into two parts for the product rule:

f(q) = q^2

g(q)= sqrt(q)-2

Product Rule: f(x)g(x)=f'(x)g(x)+f(x)g'(x) (we will sub in q for x in this case)

The product rule states that the derivative of a product of two functions

f(q) and g(q) is f'(q) g(q)+f(q) g'(q).

Differentiation Steps

Differentiate f(q)=q^2 : nx^n-1 where n=2 and n-1=1

The derivative f'(q)=2q.

Differentiate g(q)=sqrt(q)-2 : sqrt(q) is the same as q^1/2: nx^n-1 where n=1/2 and n-1=-1/2

The derivative g'(q)=1/(2sqrt(q))

Apply the Product Rule: (plug and play at this point)

C'(q)= f'(q) g(q)+f(q) g'(q)

C'(q)= (2q)(sqrt(q)-2)+(q^2)(1/(2sqrt(q)))

Simplify:

Simplify the expression to get the marginal cost.

Let's carry out these steps to find the marginal cost.

Using the f(x) and g(x) format and applying the product rule, the marginal cost MC(q) is calculated as follows:

MC(q)=(5/2)(q^3/2)-4q

I hope this helps and you can start to understand how this all works. If you have any questions feel free to book an appointment so we can go over this more in depth.

Regards,

John