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📝 Marginal cost revenue profit calculus (23 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 23 questions available

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What is Marginal cost revenue profit calculus?

Definition:
Marginal profit $P'(x) = R'(x) - C'(x)$ determines optimal production. Profit is maximized when marginal revenue equals marginal cost ( $R'(x) = C'(x)$ ). Analyzing these derivatives helps identify the production level where additional units no longer add to net profit.

Example:
If $R'(x) = 20 - 0.1x$ and $C'(x) = 5 + 0.2x$ , set $20 - 0.1x = 5 + 0.2x \Rightarrow 15 = 0.3x \Rightarrow x=50$ for max profit.

Reason:
Equating marginal rates identifies the balance point where gain from sales matches cost of production, ensuring maximum efficiency and profitability.

Easy

Medium

Hard

📝 All Marginal cost revenue profit calculus MCQs

Q1. If marginal revenue \ $R'(x) = 12\$ and marginal cost \ $C'(x) = 12\$ at \ $x = 500\$ , what can be inferred about profit at that output level?

A.Profit is at a maximum ✅

B.Profit is at a minimum

C.Profit is increasing

D.Profit is decreasing

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: When marginal revenue equals marginal cost, the additional revenue from one more unit exactly offsets the additional cost, indicating that the profit function has reached a peak. Hence the profit is maximized at that output level.

Q2. Given the linear approximation \ $P(x+1) \\approx P(x) + P'(x)\$ , what happens to profit when \ $P'(x) > 0\$ and one more unit is produced?

A.Profit decreases

B.Profit stays the same ✅

C.Profit increases

D.Cannot be determined

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A positive marginal profit means the derivative of the profit function is positive, so the linear approximation predicts an increase in profit when the output is raised by one unit. Therefore profit will increase.

Q3. If for a range of output \ $R'(x) > C'(x)\$ , what does this imply about the firm's profit trend in that range?

A.Profit is decreasing

B.Profit is constant

C.Profit is increasing ✅

D.Profit oscillates

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When marginal revenue exceeds marginal cost, each additional unit adds more revenue than cost, so total profit rises as output expands within that interval.

Q4. For the profit function \ $P(x)= -0.01x^{2}+40x-200\$ , at which output does marginal profit equal zero?

A.\

x = 1000\

B.\

x = 1500\

C.\

x = 2000\

D.\

x = 2500\

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The marginal profit is the derivative \ $P'(x) = -0.02x + 40\$ . Setting this equal to zero gives \ $x = 2000\$ . At this point the profit curve changes direction, indicating a critical point.

Q5. Two firms have marginal cost functions \ $C_{1}'(x)=5\$ and \ $C_{2}'(x)=0.01x+5\$ . If marginal revenue is \ $R'(x)=20\$ for low output, which firm enjoys greater profit?

A.Firm 1 ✅

B.Firm 2

C.Both earn equal profit

D.Not enough information

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: At low output the term \ $0.01x\$ in \ $C_{2}'\$ is negligible, making \ $C_{2}'\$ close to 5. Since both firms face the same marginal revenue, the one with the lower marginal cost (Firm 1) generates higher marginal profit, leading to greater overall profit.

Q6. Find the profit‑maximizing output when \ $R'(x)=30-0.5x\$ and \ $C'(x)=10+0.2x\$ .

A.\

x=20\

B.\

x=25\

C.\

x=28.6\

✅

D.\

x=35\

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Set marginal revenue equal to marginal cost: \ $30-0.5x = 10+0.2x\$ . Solving yields \ $0.7x = 20\$ and \ $x \\approx 28.6\$ . This is the output where profit is maximized.

Q7. Which condition guarantees that a critical point is a profit maximum?

A.\

P&#039;(x)=0\

alone

B.\

P&#039;&#039;(x)<0\

alone

C.Both \

P&#039;(x)=0\

and \

P&#039;&#039;(x)<0\

✅

D.Neither condition

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A profit maximum requires the first‑derivative test (zero slope) and the second‑derivative test (negative curvature). Only when both conditions hold can we be certain the critical point is a maximum.

Q8. Given \ $C(x)=0.02x^{2}+8x+500\$ and \ $R(x)= -0.01x^{2}+30x\$ , at what output is profit maximized?

A.\

x=300\

B.\

x=350\

C.\

x=367\

✅

D.\

x=400\

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: First compute marginal cost \ $C'(x)=0.04x+8\$ and marginal revenue \ $R'(x)= -0.02x+30\$ . Equating them gives \ $-0.02x+30 = 0.04x+8\$ → \ $0.06x = 22\$ → \ $x \\approx 367\$ . This satisfies the second‑derivative test for a maximum.

Q9. Which statement best reflects the principle that profit is maximized when marginal revenue equals marginal cost?

A.Total revenue equals total cost at the optimum

B.The additional cost of one more unit equals the additional revenue from that unit

C.Average cost equals average revenue at the optimum ✅

D.Profit is zero at the optimum

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The principle states that at the profit‑maximizing output, the extra revenue generated by an additional unit (marginal revenue) matches the extra cost of producing it (marginal cost). This condition ensures no further profit can be gained by changing output.

Q10. If a firm observes that marginal profit is decreasing while marginal revenue stays constant, what strategic decision should it consider?

A.Increase output

B.Decrease output ✅

C.Maintain current output

D.Change the product price

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A declining marginal profit indicates that each extra unit adds less profit than the previous one. With marginal revenue unchanged, the firm should reduce output to avoid producing units where marginal profit becomes negative, thereby preserving overall profitability.

Q11. Why does marginal analysis rely on the local linear approximation rather than the exact difference between \ $P(x+1)\$ and \ $P(x)\$ ?

A.Because exact differences are always zero

B.Because linear approximations are easier to compute and give accurate estimates for small changes

C.Because derivatives ignore the shape of the function

D.Because exact differences are undefined for profit functions ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The derivative provides the slope of the tangent line, offering a simple yet accurate estimate of how a function changes over a tiny interval. For one‑unit changes, this linear approximation is usually sufficient and far less cumbersome than calculating exact differences.

Q12. Suppose marginal profit is positive for \ $x<1500\$ and negative for \ $x>1500\$ , but marginal cost exceeds marginal revenue at \ $x=1500\$ . Which statement is most consistent?

A.Profit is maximized at \

x=1500\

✅

B.Profit is minimized at \

x=1500\

C.Profit is constant at \

x=1500\

D.No conclusion can be drawn

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: When marginal profit changes sign from positive to negative, the profit function reaches a peak at that point. The additional information that marginal cost exceeds marginal revenue at \ $x=1500\$ reinforces the conclusion that profit is maximized there.

Q13. At \ $x=10\$ we have \ $R'(x)=C'(x)\$ but the second derivative of profit is positive. What does this indicate?

A.Profit is at a minimum

B.Profit is at a maximum ✅

C.Profit is at an inflection point

D.Cannot determine profit level

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Equality of marginal revenue and marginal cost gives a critical point. If the second derivative of profit is positive, the profit curve is concave upward, meaning the critical point is a local minimum rather than a maximum.

Q14. If marginal profit equals zero at \ $x=0\$ and then decreases for larger \ $x\$ , what can be inferred about the shape of the profit function?

A.Profit decreases for all \

x\

B.Profit is constant then decreases

C.Profit has a maximum at \

x=0\

✅

D.Profit has a minimum at \

x=0\

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A zero marginal profit at the origin together with a negative slope thereafter means the profit function starts at a peak and then falls as output increases, indicating a maximum at \ $x=0\$ .

Q15. How does an upward shift of the marginal cost function by two units affect the profit‑maximizing output?

A.Output decreases ✅

B.Output increases

C.Output stays the same

D.Effect cannot be determined

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Raising marginal cost makes it intersect marginal revenue at a lower output level, because the cost curve reaches the revenue curve sooner. Consequently, the profit‑maximizing quantity shifts downward.

Q16. For \ $P(x)= -0.005x^{3}+0.3x^{2}+50x-1000\$ , determine the critical point(s) and identify which yields maximum profit.

A.\

x \\approx 81\

✅

B.\

x \\approx 120\

C.No maximum exists

D.Multiple maxima exist

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The derivative \ $P'(x)= -0.015x^{2}+0.6x+50\$ leads to a quadratic equation whose positive root is \ $x \\approx 81\$ . The second derivative \ $P''(x)= -0.03x+0.6\$ is negative at this point, confirming a local maximum.

Q17. In perfect competition, marginal revenue equals price. What does the condition \ $R'(x)=C'(x)\$ imply about the market price?

A.Price equals marginal cost

B.Price is greater than marginal cost

C.Price is less than marginal cost

D.Price cannot be determined ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: When a firm is a price taker, marginal revenue is the market price. Equating marginal revenue to marginal cost therefore sets the price equal to marginal cost, a hallmark of allocative efficiency in perfectly competitive markets.

Q18. What is the definition of marginal profit?

A.Total profit divided by total output

B.Additional profit from producing one more unit

C.Average profit per unit ✅

D.Difference between total revenue and total cost

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Marginal profit measures the change in total profit that results from increasing production by a single unit, formally expressed as \ $P'(x)\$ . It captures the incremental benefit of the next unit produced.

Q19. A firm finds that \ $R'(x)-C'(x)=5\$ for every \ $x\$ . What does this say about the firm's profit trend?

A.Profit increases linearly ✅

B.Profit decreases linearly

C.Profit remains constant

D.Profit oscillates

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Since the difference between marginal revenue and marginal cost is a constant positive value, each additional unit adds the same amount of profit, leading to a linear upward growth in total profit as output expands.

Q20. Determine the output level where marginal revenue \ $R'(x)=50-0.5x\$ becomes zero.

A.\

x=50\

B.\

x=75\

C.\

x=100\

✅

D.\

x=125\

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Set \ $50-0.5x = 0\$ . Solving gives \ $0.5x = 50\$ and \ $x = 100\$ . At this output, marginal revenue falls to zero, indicating no additional revenue from further units.

Q21. A multi‑product firm has positive marginal profit for product A and negative marginal profit for product B at current outputs. Which allocation strategy maximizes total profit?

A.Increase output of A only

B.Decrease output of B only ✅

C.Reallocate resources from B to A

D.Keep both outputs unchanged

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: When one product yields positive marginal profit and another yields negative marginal profit, shifting resources from the loss‑making product to the profitable one raises overall profit. The optimal move is to reallocate from B to A.

Q22. What symbol is commonly used to denote marginal cost?

A.\

C&#039;(x)\

✅

B.\

C(x)\

C.\

M_{C}\

D.\

C&#039;&#039;(x)\

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The standard notation for marginal cost, the derivative of the total cost function with respect to output, is \ $C'(x)\$ .

Q23. According to the basic economic principle, profit is maximized when which condition holds?

A.\

P&#039;(x)=0\

B.\

R&#039;(x)=C&#039;(x)\

✅

C.\

R(x)=C(x)\

D.\

P(x)=0\

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The principle states that the profit‑maximizing output occurs where marginal revenue equals marginal cost, i.e., \ $R'(x)=C'(x)\$ . This condition ensures that the extra revenue from one more unit just covers its extra cost, leaving profit unchanged for further changes.

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