The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15.

Exercise 6.1 · Q16

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Concept understanding — Marginal Revenue Calculation

Marginal Revenue: The Revenue from One More Unit

Imagine you run a small stall selling lemonade. You sell 10 glasses at ₹10 each, so total revenue is ₹100. Now you wonder: If I sell one more glass, how much extra money will I actually bring in? That extra money — the change in total revenue from the 11th glass — is marginal revenue.

The Core Intuition

Marginal revenue (MR) answers a practical question: "What happens to total revenue when I increase sales by exactly one unit?"

It is not simply the price of that extra unit. Selling one more might force you to lower the price on all units — especially if you have some market power (like a monopoly or a price-setting firm). In perfect competition, where you're a price-taker, MR equals the market price; but in most real-world scenarios, MR is less than the price.

Note

Think of MR as the slope of the total revenue curve. If total revenue is $R(q)$ , then $MR = \frac{dR}{dq}$ .

The Precise Definition

Marginal Revenue is the rate at which total revenue changes as the quantity sold changes. For a firm selling $q$ units with total revenue $R(q)$ :

MR(q) = \frac{dR}{dq}

If the demand curve is $p(q)$ (price as a function of quantity), then $R(q) = p(q)\times q$ , and by the product rule:

MR(q) = \frac{d}{dq}\big[p(q)\cdot q\big] = p(q) + q\cdot\frac{dp}{dq}

This says marginal revenue equals the price of the current unit plus the effect that selling one more has on the price of all previous units (since $dp/dq$ is usually negative — to sell more, you lower the price).

MR = p + q\cdot\frac{dp}{dq}

A Concrete Example

Suppose a monopolist faces demand $p = 100 - 2q$ (price drops ₹2 per extra unit sold), so total revenue is $R(q) = q(100-2q) = 100q - 2q^2$ .

The instantaneous marginal revenue is the derivative:

MR(q) = R'(q) = 100 - 4q

At $q = 10$ : $MR(10) = 100 - 40 = 60$ . Checking with the product-rule formula: $p(10) = 100 - 20 = 80$ , and $dp/dq = -2$ , so $MR = 80 + 10\times(-2) = 60$ — the two methods agree.

Compare this to the discrete one-unit change: at $q=10$ , $R = 80\times10 = 800$ ; at $q=11$ , $p = 78$ and $R = 78\times11 = 858$ , so the actual revenue gained from the 11th unit is $858 - 800 = 58$ .

Note

$58$ (the real jump from 10 to 11 units) and $60$ (the instantaneous derivative at exactly $q=10$ ) are close but not identical. The derivative gives the instantaneous slope of the revenue curve at $q=10$ ; the discrete difference is the average slope over the step from $q=10$ to $q=11$ , which also picks up a little of the curve's downward curvature over that step. For a smoothly varying revenue function, the two are always close but only exactly equal when $R(q)$ is linear. …

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Marginal revenue is the instantaneous rate of change of revenue with respect to quantity. For $R(x) = 13x^2 + 26x + 15$ , the marginal revenue at $x=7$ is found by differentiating and substituting, giving ₹208 per unit.

Why marginal revenue is a derivative

Marginal revenue answers a practical question: if you sell one more unit, how much extra money comes in? The function $R(x)$ gives total revenue for exactly $x$ units. But revenue doesn't jump in discrete steps — we want the rate at which revenue changes as sales increase infinitesimally. That rate is the derivative $R'(x)$ .

For a polynomial revenue function, the derivative is straightforward: each term's power drops by one and multiplies by the old exponent. Constant terms vanish because they don't change with $x$ .

Watch out

A common mistake is to plug $x=7$ into $R(x)$ first, then try to find the "difference" between $R(7)$ and $R(8)$ . That gives the average rate over one unit, not the instantaneous marginal revenue. The derivative gives the exact rate at the point itself.

Step-by-step calculation

Write down the revenue function

R(x) = 13x^2 + 26x + 15

Differentiate term by term …

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

Concept understanding — Marginal Revenue Calculation

Marginal Revenue: The Revenue from One More Unit

The Core Intuition

The Precise Definition

A Concrete Example

Why marginal revenue is a derivative

Step-by-step calculation

Method: Marginal Revenue as the Derivative of the Total Revenue Function

Steps

Common Mistakes

Q.The total revenue in Rupees received from the sale of xxx units of a product is given by R(x)=13x2+26x+15R(x) = 13x^2 + 26x + 15R(x)=13x2+26x+15. Find the marginal revenue when x=7x = 7x=7.

Why marginal revenue is a derivative

Step-by-step calculation

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

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