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Class 12 Statistics Notes · GSEB

Marginal Concepts

Marginal Concepts — compute marginal revenue, marginal cost and price elasticity of demand as derivatives, with live calculators and TR–MR curves. GSEB Class 12 Commerce Statistics notes.

Last updated: 24 Sep 2026

Notes

Marginal Concepts — The Business Face of Derivatives

Marginal means the change caused by one extra unit. In calculus language it is simply the first derivative of the relevant function — this is where differentiation starts paying rent in business.

Marginal Revenue (MR)

Marginal Revenue
The change in revenue due to a small change in demand — the extra rupees earned from selling one more unit. MR = dR/dx.

★ Marginal Revenue

MR=dRdx,R=p⋅xMR = \frac{dR}{dx}, \quad R = p \cdot x

If demand function is p = f(x), first build R = p·x, then differentiate with respect to x.

Real reading — the pizza example

p = 150 − 4x gives R = 150x − 4x², so MR = 150 − 8x. At x = 3, MR = ₹126 — the 4th pizza adds about ₹126 to revenue. Not the price (₹138) — the extra revenue, because selling one more forces the price down on ALL units.

TR and MR Curves — Build Your Own

Start with the pizza stall (p = 150 − 4x), or type your own demand function. The lab builds R = x·p, differentiates to get MR, and marks where MR crosses zero — the revenue summit.

TR / MR Curve Lab — demand p(x) = −4x + 150
Try your own demand function p(x)

The lab builds R(x) = x·p(x), differentiates it to get MR, draws both curves, and marks where MR = 0 (maximum revenue).

R(x) = x · p(x) = −4x² + 150x

MR(x) = dR/dx = −8x + 150

Total Revenue vs Marginal Revenue — p = −4x + 150

02.557.510-102304697099481188Quantity x₹

No MR = 0 in range

MR(−8x + 150) never reaches zero for x ≥ 0 — revenue keeps rising (or falling) across the plotted range. Check that your demand function actually slopes downward (negative x term).

Live Marginal Calculators

Marginal Revenue — p = 150 − 4x

MR=150−8xMR = 150 − 8x

Marginal Revenue

₹ 126

Total Revenue R = px

₹ 414

Price p

₹ 138

Marginal Cost — C = 5x² + 6x + 2000

MC=10x+6MC = 10x + 6

Marginal Cost

₹ 506

Total Cost C

₹ 14,800

Elasticity — x = 50 − 4p

e=4p/(50−4p)e = 4p / (50 − 4p)

Elasticity |e|

0.67

Demand x

30

Verdict

-1

Verdict key: 1 = elastic, 0 = unit, −1 = inelastic. At p = 5, e = 20/30 ≈ 0.67 → inelastic (matches the textbook illustration).

Formula Sheet

Marginal concepts at a glance
ConceptFormulaReads as
Total RevenueR = p · xprice × quantity
Marginal RevenueMR = dR/dxextra ₹ from one more unit sold
Total CostC = FC + VCfixed + variable cost
Marginal CostMC = dC/dxextra ₹ to produce one more unit
Elasticitye = −(p/x)(dx/dp)% change in demand ÷ % change in price

Solved Examples

Solved Example

Problem

Demand function of pizza: p = 150 − 4x. Find marginal revenue when demand is 3 pizzas and interpret.

Solution

MR = ₹126 — revenue from selling the 4th pizza is about ₹126

Solved Example

Problem

Cost function C = 5x² + 6x + 2000. Find marginal cost at 50 units and interpret.

Solution

MC = ₹506 — producing the 51st unit costs about ₹506

Solved Example

Problem

Demand x = 50 − 4p. Find elasticity at p = 5 and interpret.

Solution

e ≈ 0.67 — a 1% price change moves demand by about 0.67% (inelastic at this price)

Solved Example

Problem

Find marginal revenue if revenue function is R = 90x − x²/2. (Section C)

Solution

MR = 90 − x

Key Takeaways

Key Takeaways

  • ★ MR = dR/dx and MC = dC/dx — marginal = derivative = extra unit effect.
  • Build R = p·x first when given a demand function, then differentiate; fixed costs drop out of MC automatically.
  • ★ Elasticity e = −(p/x)(dx/dp): |e| > 1 elastic, |e| < 1 inelastic, |e| = 1 unit (where revenue peaks).
  • MR = 0 marks maximum total revenue; beyond it, extra sales destroy revenue.
  • So what? Every pricing decision — Swiggy discount depth, kirana bulk offers, airline fares — is an elasticity and marginal-revenue judgement.

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