Question

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R\left( x \right) = 13{x^2} + 26x + 15$. Find the marginal revenue when $x = 7$.

Answer 1

Hint: First find the formula of marginal revenue as function of the units produced by differentiating the given revenue formula. Then use the obtained function to find the marginal revenue at the given units.

Complete step-by-step solution -

Revenue is the amount of income that a company receives from the sale of products or services in a particular period of time.
Marginal revenue is the additional revenue that will be generated by increasing product sales by one unit.
Hence marginal revenue would be given by change in revenue per unit change in sales of units.
Hence if the revenue is given a function $f(x)$, then the marginal revenue is given by its derivative, that is $\dfrac{{d\left( {f(x)} \right)}}{{dx}}$.
Given the problem, total revenue in Rupees received from the sale of $x$ units of a product is given by:
$R\left( x \right) = 13{x^2} + 26x + 15{\text{ (1)}}$
We need to find the marginal revenue when $x = 7$.
Hence as per the above definition, we need to find the derivative of the equation (1).
The marginal revenue $M(x)$is given by:
$M\left( x \right) = \dfrac{{d\left( {R\left( x \right)} \right)}}{{dx}} = \dfrac{{d\left( {13{x^2} + 26x + 15} \right)}}{{dx}}{\text{ (2)}}$
We know that the differentiation of a monomial $a{x^n}$is given by $\dfrac{{d\left( {a{x^n}} \right)}}{{dx}} = na{x^{n - 1}}$.
And also $\dfrac{{d\left( {f(x) + g(x)} \right)}}{{dx}} = \dfrac{{d\left( {f(x)} \right)}}{{dx}} + \dfrac{{d\left( {g(x)} \right)}}{{dx}}$.
Using the above results in equation (2), we get
\[
  M\left( x \right) = \dfrac{{d\left( {13{x^2} + 26x + 15} \right)}}{{dx}} = \dfrac{{d\left( {13{x^2}} \right)}}{{dx}} + \dfrac{{d\left( {26x} \right)}}{{dx}} + \dfrac{{d\left( {15} \right)}}{{dx}} \\
   \Rightarrow M\left( x \right) = 13 \times 2{x^{2 - 1}} + 1 \times 26{x^{1 - 1}} + 0 \\
   \Rightarrow M\left( x \right) = 26x + 26 \\
\]
Hence the marginal revenue is given by
\[M\left( x \right) = 26x + 26{\text{ (3)}}\]
We need to find the marginal revenue when $x = 7$.
Marginal revenue at $x = 7$ is given by,
\[M\left( 7 \right) = 26 \times 7 + 26 = 208\]
Hence the marginal revenue when $x = 7$ is equal to Rs. 208.

Note: The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. The relationship of the revenue and the marginal revenue should be kept in mind while solving problems like above. The differentiation formulas should be used directly in problems like above. The differentiation should be done with respect to the same variable to which the result “as a function of” is desired.