MHB From the marginal cost to the total cost.

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The discussion focuses on deriving the total cost (TC) from the marginal cost (MC) function for two companies. The marginal cost is defined as $MC_{i}(q)_{i}=q_{i}+10$, and the goal is to integrate this to find the total cost. By applying integration and the Fundamental Theorem of Calculus, it is established that with no fixed costs, the total cost can be expressed as $TC=\frac{1}{2}q^2+10q$. The parameters used in the integration confirm that the coefficients align with the given marginal cost function. This process illustrates the relationship between marginal and total costs in economic analysis.
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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 $C_T$:

$$\d{C_T}{q}=C_M$$

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:

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

Applying the FTOC, there results:

$$C_T-C_F=\frac{a}{2}q^2+bq$$

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

$$C_T=\frac{1}{2}q^2+10q$$
 
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