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📝 Marginal analysis calculus economics (24 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 24 questions available

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What is Marginal analysis calculus economics?

Definition:
Marginal analysis uses derivatives to estimate the change in cost, revenue, or profit for producing one additional unit. Marginal cost $C'(x)$ , revenue $R'(x)$ , and profit $P'(x)$ approximate instantaneous rates of change, aiding decision-making in production and pricing strategies.

Example:
If $C(x) = 100 + 5x + 0.1x^2$ , marginal cost $C'(x) = 5 + 0.2x$ . At $x=10$ , $C'(10) = 7$ , meaning the 11th unit costs approx $\$ 7$.

Reason:
Derivatives provide quick approximations for incremental changes, allowing businesses to optimize production levels without recalculating total costs for each unit.

Easy

Medium

Hard

📝 All Marginal analysis calculus economics MCQs

Q1. Which of the following correctly defines the profit function $P(x)$ in terms of revenue $R(x)$ and cost $C(x)$ ?

P(x)=C(x)-R(x)

P(x)=R(x)+C(x)

P(x)=R(x)-C(x)

✅

P(x)=R(x)\times C(x)

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Profit is calculated as revenue minus cost, so the correct expression is $P(x)=R(x)-C(x)$ . The other choices either reverse the order or combine the quantities incorrectly.

Q2. In the cost function $C(x)=a+bx+cx^{2}$ , what does the constant $a$ represent?

A.Variable manufacturing cost per unit

B.Fixed overhead cost independent of output ✅

C.Quadratic cost coefficient

D.Price per unit

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The constant $a$ captures overhead costs such as rent and insurance that must be paid regardless of how many units are produced.

Q3. If each unit sells for $p$ dollars, which expression gives the total revenue for $x$ units?

R(x)=p+x

R(x)=px

✅

R(x)=p-x

R(x)=p x^{2}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Revenue equals price per unit times quantity sold, so $R(x)=px$ . The other formulas do not represent total sales revenue.

Q4. Given a production capacity limit $l$ , the feasible values of $x$ satisfy which interval?

0 < x < l

-l \le x \le l

0 \le x \le l

✅

x \ge l

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The firm can produce any non‑negative amount up to its capacity, so the admissible interval is $0 \le x \le l$ .

Q5. For the profit function $P(x)=200x-(500000+80x+0.003x^{2})$ , what is the derivative $P'(x)$ equal to?

200-80-0.006x

✅

200-80+0.006x

200+80-0.006x

200-0.003x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Differentiating gives $P'(x)=200-(80+0.006x)=200-80-0.006x$ . This represents marginal profit.

Q6. Which cost function has a higher fixed overhead than $C(x)=500000+80x+0.003x^{2}$ ?

C(x)=600000+80x+0.003x^{2}

✅

C(x)=500000+70x+0.003x^{2}

C(x)=500000+80x+0.002x^{2}

C(x)=500000+80x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The first option adds \$100,000 to the constant term, raising the overhead while leaving the variable parts unchanged.

Q7. If the selling price per unit increases from \ $200 to \$ 250, what is the immediate effect on the profit function $P(x)=px-(a+bx+cx^{2})$ ?

A.The linear term coefficient increases, shifting the profit curve upward. ✅

B.The quadratic term coefficient increases.

C.The overhead

a

decreases.

D.The profit function becomes independent of

x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Raising $p$ raises the coefficient of the linear term $px$ , lifting the entire profit curve without altering its shape.

Q8. If the overhead $a$ rises while $b, c,$ and $p$ stay unchanged, how does the profit‑maximizing output $x^{*}$ change?

x^{*}

increases.

x^{*}

decreases.

x^{*}

stays the same. ✅

D.Cannot determine without numbers.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The optimal output is found from $P'(x)=p-b-2cx=0$ , which does not involve $a$ . Therefore changes in overhead do not affect the maximizing $x$ .

Q9. Consider $P_{1}(x)=200x-(500000+80x+0.003x^{2})$ and $P_{2}(x)=210x-(500000+90x+0.003x^{2})$ . Which statement is true about their profit‑maximizing outputs?

x^{*}_{1}=x^{*}_{2}

✅

x^{*}_{1}>x^{*}_{2}

x^{*}_{1}<x^{*}_{2}

D.No maximum exists for either.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Both give $x^{*}=(p-b)/(2c)$ . For each, $p-b=200-80=120$ and $210-90=120$ ; with the same $c$ , the optimal $x$ is identical.

Q10. Using $P(x)=200x-(500000+80x+0.003x^{2})$ , what is the profit‑maximizing production level $x^{*}$ (ignore capacity limit)?

A.10,000 units

B.15,000 units

C.20,000 units ✅

D.25,000 units

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Set $P'(x)=0$ : $200-80-0.006x=0$ ⇒ $x=120/0.006=20,000$ .

Q11. What is the marginal profit $P'(x)$ when $x=10,000$ for the profit function in the example?

A.\$60 ✅

B.\$80

C.\$100

D.\$120

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: $P'(x)=200-80-0.006x$ . Substituting $x=10,000$ gives $120-60=60$ dollars.

Q12. If the optimal $x^{*}=20,000$ exceeds the capacity limit $l=15,000$ , what production level will the firm actually choose to maximize profit?

A.0

B.15000 ✅

C.20000

D.30000

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When the unconstrained optimum lies beyond capacity, the firm produces at the highest feasible level, i.e., the capacity limit $l=15,000$ .

Q13. In the cost function $C(x)=a+bx+cx^{2}$ , a positive $c$ indicates what about marginal cost as output increases?

A.Marginal cost decreases.

B.Marginal cost remains constant.

C.Marginal cost increases. ✅

D.Marginal cost becomes negative.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The marginal cost is $C'(x)=b+2cx$ ; with $c>0$ , this derivative grows as $x$ grows, so marginal cost rises.

Q14. If the selling price $p$ increases by 10%, how does the optimal output $x^{*}=\frac{p-b}{2c}$ change, assuming $b$ and $c$ are unchanged?

A.Increases by 10%

B.Increases by less than 10% ✅

C.Decreases

D.Remains unchanged

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because $x^{*}$ is proportional to the numerator $p-b$ , a 10% rise in $p$ raises $x^{*}$ but not by the full 10% unless $b=0$ .

Q15. Given the capacity limit of 30,000 units, what is the profit $P(30,000)$ for the example function?

A.\$400,000 ✅

B.\$500,000

C.\$300,000

D.\$600,000

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: $P(30,000)=200(30,000)-(500,000+80(30,000)+0.003(30,000)^2)=6,000,000-5,600,000=400,000$ .

Q16. The break‑even output occurs when profit equals zero. For the example, which equation must be solved to find the break‑even points?

200x = 500000+80x+0.003x^{2}

✅

200x = 0

500000+80x+0.003x^{2}=0

200 = 500000+80+0.003x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Setting $P(x)=0$ gives $200x-(500000+80x+0.003x^{2})=0$ , which simplifies to the listed equation.

Q17. If the quadratic cost coefficient $c$ is reduced, what is the expected effect on the shape of the profit function $P(x)=px-(a+bx+cx^{2})$ ?

A.The profit curve becomes more concave (steeper downwards).

B.The profit curve becomes less concave (flatter). ✅

C.The profit curve becomes convex.

D.No change.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A smaller $c$ reduces the magnitude of the negative quadratic term, making the profit curve flatter (less sharply concave).

Q18. For $P(x)=200x-(500000+80x+0.003x^{2})$ with capacity $l=30,000$ , what is the profit‑maximizing output $x^{*}$ and corresponding profit?

x^{*}=20,000,\; P=200,000

✅

x^{*}=20,000,\; P=400,000

x^{*}=30,000,\; P=400,000

x^{*}=15,000,\; P=600,000

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Setting $P'(x)=0$ gives $x^{*}=20,000$ . Substituting back yields profit $200(20,000)-[500,000+80(20,000)+0.003(20,000)^2]=200,000$ .

Q19. If the linear cost coefficient $b$ increases by 20 while the price $p$ stays at 200, how does the optimal output $x^{*}=\frac{p-b}{2c}$ change?

A.Decreases by

\frac{20}{2c}

✅

B.Increases by

\frac{20}{2c}

C.Remains unchanged

D.Becomes negative

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: An increase in $b$ reduces the numerator $p-b$ ; the optimal output falls by exactly $\frac{20}{2c}$ .

Q20. Given that the profit function $P(x)=px-(a+bx+cx^{2})$ is concave downwards for all $x$ , what must be true about the coefficient $c$ ?

c>0

✅

c<0

c=0

D.No restriction on

c

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The second derivative is $P''(x)=-2c$ . For the function to be concave (negative second derivative), $-2c<0$ ⇒ $c>0$ .

Q21. Why does the profit function $P(x)=px-(a+bx+cx^{2})$ with $c>0$ guarantee a single global maximum on $[0,l]$ ?

A.Because

P(x)

is a linear function.

B.Because the second derivative

P&#039;&#039;(x)=-2c

is negative, making

P

strictly concave. ✅

C.Because the first derivative is always positive.

D.Because the cost function dominates revenue.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A negative constant second derivative means the function is strictly concave, so any critical point is the unique global maximum within the interval.

Q22. If the overhead $a$ increases by \$100,000, how does the maximum profit change, assuming the optimal output remains at $x^{*}=20,000$ ?

A.Decreases by \$100,000 ✅

B.Increases by \$100,000

C.Remains unchanged

D.Decreases by less than \$100,000

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Overhead appears additively in the profit expression, so a \$100,000 rise reduces profit by the same amount, regardless of output level.

Q23. A firm wants a profit of at least \$500,000 with price $p=200$ and capacity $l=30,000$ . Which choice of overhead $a$ (keeping $b=80, c=0.003$ ) ensures this target is met at the optimal output?

A.\$500,000

B.\$600,000

C.\$700,000 ✅

D.\$800,000

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: At the optimal $x=20,000$ , profit equals $1,200,000-a$ . To achieve ≥\ $' in math mode at position 10: 500,000, \̲(̲a$ must be ≤\" style="color:#cc0000">500,000, $a$ must be ≤\700,000; the highest permissible choice listed is \$700,000.

Q24. Given $p=200$ , $b=80$ , and $c=0.003$ , which of the following best describes the behavior of profit as $x$ approaches the capacity limit 30,000?

A.Profit continues to increase beyond 30,000.

B.Profit reaches its maximum at

x=20,000

then declines. ✅

C.Profit is constant for all

x

in [0,30,000].

D.Profit is negative for all

x

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The profit function is concave with a peak at $x^{*}=20,000$ ; beyond this point, additional units reduce profit, so at the capacity limit profit has already begun to fall.

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