From the marginal cost to the total cost.

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In summary, the conversation is about solving for the total cost given the marginal cost and quantity for two companies. The formula for total cost is derived using the definition of marginal cost and integrating both sides. In this specific problem, there are no fixed costs and the values for a and b are given as 1 and 10 respectively. The resulting formula for total cost is C_T = $\frac{1}{2}q^2+10q$.
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Francobati
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Hello. Can you help me figure out how to pass, integrating, by the marginal cost: $MC_{i}(q)_{i}=q_{i}+10$ to the total cost: $TC=\frac{1} {2}q_i^2+10q_{i}$?
$i=1,2$, are the two companies. $q_{i}$ is the quantity. What are the calculations?
 
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Given a marginal cost $C_M$, a fixed cost $C_F$ and a quantity $q$, we are to assume (using the definition of marginal cost) for the total cost $C_T$:

\(\displaystyle \d{C_T}{q}=C_M\)

Now, if we integrate both sides w.r.t $q$, exchange the dummy variables of integration (and use a linear marginal cost function) and using the given boundaries, we obtain:

\(\displaystyle \int_{C_F}^{C_T}\,du=\int_0^q av+b\,dv\)

Applying the FTOC, there results:

\(\displaystyle C_T-C_F=\frac{a}{2}q^2+bq\)

Now, for this problem, it would appear there are no fixed costs ($C_F=0$), and we are given $(a,b)=(1,10)$, hence:

\(\displaystyle C_T=\frac{1}{2}q^2+10q\)
 

FAQ: From the marginal cost to the total cost.

What is the marginal cost?

The marginal cost is the additional cost incurred by producing one more unit of a good or service. It takes into account the change in total cost divided by the change in quantity.

How is marginal cost calculated?

To calculate marginal cost, you divide the change in total cost by the change in quantity. This gives you the cost of producing one additional unit.

What is the relationship between marginal cost and total cost?

Marginal cost is a component of total cost. It represents the cost of producing one more unit, while total cost includes all costs incurred in producing a certain quantity of goods or services.

Why is it important to understand the concept of marginal cost?

Understanding marginal cost is important for businesses to determine the most efficient level of production. It can also help in pricing decisions and maximizing profits.

How does marginal cost affect the supply curve?

Marginal cost is a key factor in determining the shape of the supply curve. As marginal cost increases, the supply curve shifts upwards, indicating that more units of the good or service will only be produced at a higher cost.

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