Q.The demand function for a commodity is $p = 100 - 2x$, where $p$ is the price per unit and $x$ is the quantity demanded. Find the total revenue function and the marginal revenue when $x=10$.

Marginal cost is the rate of change of total cost with respect to output: $MC = \dfrac{dC}{dx}$, i.e. the cost of producing one additional unit.

GSEB Class-12 Statistics, Differentiation (business applications):

Marginal cost is the approximate change in total cost when the level of production is increased by one unit. If the total cost of producing $x$ units is $C = C(x)$, then marginal cost is the derivative of the total cost function:

$MC = \frac{dC}{dx}$

It tells a producer how much extra cost is added by making one more unit, which is essential for output and pricing decisions.

For $y=x^3+19$, the marginal value equals $27$ when $x=\pm 3$.

The marginal value is the first derivative of the function:

$\frac{dy}{dx}=\frac{d}{dx}\left(x^3+19\right)=3x^2.$

Set the marginal value equal to $27$:

$3x^2=27\ \Rightarrow\ x^2=9\ \Rightarrow\ x=\pm 3.$

From $MR=AR\left(1-\tfrac1{\eta_d}\right)$, solving for $\eta_d$ yields $\eta_d=\dfrac{AR}{AR-MR}$.

The standard relationship between marginal revenue $(MR)$, average revenue $(AR)$ and price elasticity of demand $(\eta_d)$ is

$MR = AR\left(1 - \frac{1}{\eta_d}\right)$

Rearrange to isolate $\eta_d$:

$\frac{MR}{AR} = 1 - \frac{1}{\eta_d} \;\Rightarrow\; \frac{1}{\eta_d} = 1 - \frac{MR}{AR} = \frac{AR - MR}{AR}$

Taking the reciprocal,

$\eta_d = \frac{AR}{AR - MR}$