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📝 Consequences Of The Mean Value Theorem

📖 From Calculus • 5. The derivative in graphing and applications • 26 questions available

Practice MCQs for Consequences Of The Mean Value Theorem. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-07-15

8
Easy Questions
12
Medium Questions
6
Hard Questions

📝 Sample Questions

Q1. A car travels from the start line at \(x=0\) km to the finish line at \(x=100\) km in exactly 2 hours. Its position function \(s(t)\) is continuous on \([0,2]\) and differentiable on \((0,2)\). According to the Mean Value Theorem, which of the following must be true?

🔹 A. A) There is a time when the car's speed is 0 km/h.
🔹 B. B) There is a time when the car's speed is 25 km/h.
🔹 C. C) There is a time when the car's speed is 50 km/h.
🔹 D. D) There is a time when the car's speed is 100 km/h

💡 Difficulty: easy | ✅ Correct: C

Q2. Using the Mean Value Theorem, which of the following best justifies the inequality \(\sin x \le x\) for \(x>0\)?

🔹 A. A) \(\sin x\) is always less than its tangent line at the origin.
🔹 B. B) The derivative of \(\sin x\) is always greater than 1.
🔹 C. C) The average rate of change of \(\sin x\) on \([0,x]\) cannot exceed 1.
🔹 D. D) \(\sin x\) reaches its maximum at \(x=\pi/2\).

💡 Difficulty: medium | ✅ Correct: C

Q3. If a function satisfies \(f'(x)\ge m>0\) for every \(x\) in \([a,b]\), what growth condition does the Mean Value Theorem imply?

🔹 A. A) \(f(b)-f(a)\le m(b-a)\).
🔹 B. B) \(f(b)-f(a)=m(b-a)\).
🔹 C. C) \(f(b)-f(a)\ge m(b-a)\).
🔹 D. D) No relationship can be deduced.

💡 Difficulty: hard | ✅ Correct: C

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🔗 Related Topics

📝 Approximating Roots📝 Areas and Limits📝 Continuity in Applications📝 Continuity of Compositions📝 Continuity of Inverse Functions
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