Q.The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x)=0.007x3−0.003x2+15x+4000. Find the marginal cost when 17 units are produced.
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Marginal Cost: The Cost of One More
Imagine you run a small bakery. Every morning you bake 50 loaves; your total cost — rent, flour, yeast, electricity, your own time — comes to ₹2,000, an average of ₹40 per loaf.
Now a customer asks for one more loaf. What does baking that 51st loaf actually cost? Not ₹40 — the rent doesn't change and your time is already paid; the oven is already hot. You need only a little more flour, yeast, and electricity: maybe ₹15.
That ₹15 is the marginal cost.
The Core Intuition
Marginal cost answers one question: "If I produce one more unit, how much does my total cost increase?"
It is not the average cost. It is not the total cost. It is the change in total cost when output changes by exactly one unit.
Think of it as the "extra cost" — the cost of the next step, not the cost of all steps so far.
The Precise Definition
Let TC(Q) be the total cost of producing Q units. The marginal cost of the Q-th unit — of increasing output from Q−1 to Q — is:
MC(Q)=TC(Q)−TC(Q−1)
When output can change continuously, we use the derivative:
MC(Q)=dQdTC
MC=ΔQΔTCorMC=dQdTC
Why It Matters
Marginal cost is the decision-maker's cost. When a firm asks "Should I produce one more unit?", the answer depends on whether the marginal revenue exceeds the marginal cost. If yes, produce it; if no, stop.
This is why marginal cost typically:
- Falls initially — fixed costs are spread out and workers specialise.
- Rises eventually — because of diminishing returns (more workers in a fixed kitchen get in each other's way).
A common mistake: confusing marginal cost with average cost. If average cost is ₹40 and marginal cost is ₹15, producing one more lowers the average — but the decision is still based on marginal cost, not average.
A Quick Example
| Loaves | Total Cost (₹) | Marginal Cost (₹) |
|---|---|---|
| 0 | 500 | — |
| 1 | 520 | 20 |
Concept: Marginal Cost — the rate of change of total cost with respect to output, given by the derivative C′(x).
Step 1: Differentiate C(x).
C′(x)=0.021x2−0.006x+15
Step 2: Substitute x=17.
C′(17)=0.021(289)−0.006(17)+15
Step 3: Compute. …
Marginal cost is C′(x)=0.021x2−0.006x+15. At x=17, C′(17)=6.069−0.102+15=20.967 Rupees.
Solution
1. Marginal cost is the derivative of total cost.
C(x)=0.007x3−0.003x2+15x+4000.
2. Differentiate term by term (the constant 4000 vanishes):
C′(x)=3(0.007)x2−2(0.003)x+15=0.021x2−0.006x+15.
3. Evaluate at x=17 (using 172=289): …
Method: Marginal Cost as the Derivative of the Total Cost Function
This method applies to any question asking for the marginal cost (or marginal value of any economic total-function) at a given production level — the derivative evaluated at a point, applied to a cost polynomial.
Steps
Step 1: Recognise that "marginal cost" means the derivative of the total cost function
Marginal cost is the instantaneous rate at which total cost changes as output increases — that is, MC(x)=C′(x), not C(x) itself and not the average cost xC(x).
Step 2: Differentiate the total cost function term by term
Apply the power rule to each term of C(x); a constant term (fixed cost) differentiates to zero, since fixed costs do not change as one more unit is produced.
Step 3: Write down the marginal cost function C′(x)
This is a new function of x — valid for finding the marginal cost at ANY production level, not just the one asked about.
Step 4: Substitute the specific value of x given in the question …
Common Mistakes
Mistake 1: Computing average cost instead of marginal cost
Why it's wrong: some students compute 17C(17) (the average cost per unit at 17 units) instead of C′(17) (the derivative) — these are two genuinely different economic quantities, and marginal cost is specifically about the derivative, the cost of producing roughly one MORE unit, not the average cost so far. Correct approach: always differentiate C(x) first to get C′(x), then evaluate at the given x — never divide C(x) by x for a "marginal" question.
Mistake 2: Sign or coefficient slip on the −0.003x2 term …
- CA Foundation 2026Set may-20261 markMCQQ.Use the following information for Q 59, Q 60 and Q 61;Calculate the marginal cost of the 5th unit of production. (A) ₹ 8 (B) ₹ 23.60 (C) ₹ 43.60 (D) ₹ 118
No. of units Total Fixed Cost (₹) Total Variable Cost (₹) 0 100 0 1 100 50 2 100 80 3 100 100 4 100 110 5 100 118 ›Reveal solutionSolution
MC of the 5th unit = TVC₅ − TVC₄ = 118 − 110 = ₹8.
Step 1 — Marginal cost formula
Marginal cost = change in total cost when output rises by one unit. Since fixed cost is constant, MC=ΔTVC.
Step 2 — Read the relevant figures
No. of units Total Fixed Cost (₹) Total Variable Cost (₹) 4 100 110 5 100 118 Step 3 — Compute
MC5=TVC5−TVC4=118−110=₹8
(Equivalently, TC₅ − TC₄ = 218 − 210 = ₹8.) …
- CA Foundation 2025Set may-20251 markMCQQ.Use the following data to answer question 33 and 34 :Between 10 and 20 units, what is the marginal cost per unit ? (A) ₹ 10 (B) ₹ 20 (C) ₹ 100 (D) ₹ 220
Quantity 0 10 20 30 40 Total Cost (in ₹) 100 220 320 410 510 ›Reveal solutionSolution
MC per unit = ΔTC ÷ ΔQ = (320 − 220) ÷ (20 − 10) = 100/10 = ₹10.
Step 1 — Change in total cost
Between 10 and 20 units:
ΔTC=320−220=₹100
Step 2 — Change in output
ΔQ=20−10=10 units
Step 3 — Marginal cost per unit
MC=ΔQΔTC=10100=₹10
Why the other options are wrong
- (B) ₹20 doubles the correct figure.
- (C) ₹100 is the TOTAL extra cost of the batch, not the PER-UNIT marginal cost.
- (D) ₹220 is simply the total cost at 10 units — not marginal at all. …
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