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Differentiability

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Q. If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(2).
  • A. 0
  • B. 1
  • C. 2
  • D. 3
Q. If f(x) = x^4 - 4x^3 + 6x^2, find f'(2).
  • A. 0
  • B. 1
  • C. 2
  • D. 3
Q. If f(x) = { x^2, x < 0; kx + 1, x >= 0 } is differentiable at x = 0, what is k?
  • A. -1
  • B. 0
  • C. 1
  • D. 2
Q. If f(x) = |x|, is f differentiable at x = 0?
  • A. Yes
  • B. No
  • C. Only from the right
  • D. Only from the left
Q. Is the function f(x) = x^2 - 2x + 1 differentiable at x = 1?
  • A. Yes
  • B. No
  • C. Only from the left
  • D. Only from the right
Q. Is the function f(x) = x^2 - 4x + 4 differentiable at x = 2?
  • A. Yes
  • B. No
  • C. Only from the left
  • D. Only from the right
Q. Is the function f(x) = x^2 - 4x + 4 differentiable everywhere?
  • A. Yes
  • B. No
  • C. Only at x = 0
  • D. Only at x = 2
Q. Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?
  • A. Yes
  • B. No
  • C. Only from the left
  • D. Only from the right
Q. Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
  • A. Yes
  • B. No
  • C. Only left differentiable
  • D. Only right differentiable
Q. The function f(x) = e^x is differentiable at all points?
  • A. True
  • B. False
  • C. Only at x = 0
  • D. Only at x = 1
Q. The function f(x) = ln(x) is differentiable at x = 1?
  • A. Yes
  • B. No
  • C. Only for x > 1
  • D. Only for x < 1
Q. The function f(x) = sqrt(x) is differentiable at x = 0?
  • A. Yes
  • B. No
  • C. Only from the right
  • D. Only from the left
Q. The function f(x) = x^2 + 2x + 1 is differentiable everywhere?
  • A. True
  • B. False
  • C. Only at x = 0
  • D. Only for x > 0
Q. The function f(x) = x^2 - 2x + 1 is differentiable at all points?
  • A. True
  • B. False
  • C. Only at x = 0
  • D. Only for x > 0
Q. The function f(x) = x^2 - 2x + 1 is differentiable at x = 2?
  • A. Yes
  • B. No
  • C. Only left
  • D. Only right
Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
  • A. Yes
  • B. No
  • C. Only left
  • D. Only right
Q. The function f(x) = x^2 - 4x + 4 is differentiable everywhere?
  • A. True
  • B. False
  • C. Only at x = 0
  • D. Only at x = 2
Q. The function f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1 is differentiable at x = 1?
  • A. Yes
  • B. No
  • C. Only continuous
  • D. Only from the left
Q. The function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0 is differentiable at x = 0. True or False?
  • A. True
  • B. False
  • C. Depends on x
  • D. Not enough information
Q. The function f(x) = x^3 - 3x + 2 is differentiable at x = 1?
  • A. Yes
  • B. No
  • C. Only left
  • D. Only right
Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. Find its critical points.
  • A. -1
  • B. 0
  • C. 1
  • D. 2
Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. What is f'(1)?
  • A. 0
  • B. 1
  • C. 2
  • D. 3
Q. The function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } is differentiable at x = 1 if which condition holds?
  • A. f(1) = 1
  • B. f'(1) = 1
  • C. f'(1) = 2
  • D. f(1) = 2
Q. The function f(x) = |x| is differentiable at x = 0?
  • A. Yes
  • B. No
  • C. Only from the right
  • D. Only from the left
Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
  • A. 0
  • B. 1
  • C. 2
  • D. undefined
Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 1?
  • A. 0
  • B. 1
  • C. 1/2
  • D. 1/3
Q. What is the derivative of f(x) = sin(x^2)?
  • A. 2x cos(x^2)
  • B. cos(x^2)
  • C. 2x sin(x^2)
  • D. sin(x^2)
Q. What is the derivative of f(x) = |x| at x = 0?
  • A. 0
  • B. 1
  • C. -1
  • D. Undefined
Q. Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }
  • A. f(1) = 1
  • B. f(1) = 0
  • C. f(1) = 2
  • D. f(1) = 3
Q. Which of the following functions is differentiable everywhere?
  • A. f(x) =
  • B. x
  • C.
  • D. f(x) = x^2
  • . f(x) = sqrt(x)
  • . f(x) = 1/x
Showing 61 to 90 of 91 (4 Pages)

Differentiability MCQ & Objective Questions

Differentiability is a crucial concept in calculus that plays a significant role in various examinations. Understanding this topic not only helps in grasping advanced mathematical concepts but also enhances your problem-solving skills. Practicing MCQs and objective questions on differentiability is essential for scoring better in your exams. By solving these practice questions, you can identify important questions and strengthen your exam preparation.

What You Will Practise Here

  • Definition and significance of differentiability
  • Conditions for differentiability at a point
  • Relationship between continuity and differentiability
  • Derivatives and their applications
  • Higher-order derivatives
  • Graphical interpretation of differentiable functions
  • Common differentiability problems and solutions

Exam Relevance

The topic of differentiability is frequently tested in CBSE, State Boards, NEET, and JEE examinations. Students can expect questions that require them to determine whether a function is differentiable at a given point or to apply the concept of derivatives in real-world scenarios. Common question patterns include multiple-choice questions that assess both theoretical understanding and practical application of differentiability concepts.

Common Mistakes Students Make

  • Confusing differentiability with continuity
  • Overlooking the conditions required for a function to be differentiable
  • Misinterpreting the graphical representation of differentiable functions
  • Neglecting higher-order derivatives in complex problems
  • Failing to apply the correct formulas in objective questions

FAQs

Question: What is the difference between continuity and differentiability?
Answer: Continuity means a function does not have any breaks or jumps, while differentiability indicates that a function has a defined derivative at a point.

Question: How can I determine if a function is differentiable at a point?
Answer: A function is differentiable at a point if it is continuous at that point and the limit of the difference quotient exists.

Question: Why is practicing MCQs on differentiability important?
Answer: Practicing MCQs helps reinforce your understanding, improves problem-solving speed, and prepares you for the types of questions you will encounter in exams.

Now is the time to enhance your understanding of differentiability! Dive into our practice MCQs and test your knowledge to ensure you are well-prepared for your upcoming exams.

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