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
Show solution
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer:
A
— True
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Q. The function f(x) = x^2 - 2x + 1 is differentiable at x = 2?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 2.
Correct Answer:
A
— Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence differentiable everywhere including at x = 2.
Correct Answer:
A
— Yes
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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
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere.
Correct Answer:
A
— True
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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
Show solution
Solution
f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.
Correct Answer:
B
— No
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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
Show solution
Solution
True, as the limit of f'(x) as x approaches 0 exists and equals 0.
Correct Answer:
A
— True
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Q. The function f(x) = x^3 - 3x + 2 is differentiable at x = 1?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 1.
Correct Answer:
A
— Yes
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. Find its critical points.
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Solution
f'(x) = 3x^2 - 3 = 0 gives x = ±1, thus critical points are x = -1 and x = 1.
Correct Answer:
B
— 0
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. What is f'(1)?
Show solution
Solution
f'(x) = 3x^2 - 3, thus f'(1) = 0.
Correct Answer:
A
— 0
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Q. The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Show solution
Solution
Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Correct Answer:
B
— 1
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Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer:
B
— No
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Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
A.
Continuous at x = 0
B.
Not continuous at x = 0
C.
Continuous everywhere
D.
None of the above
Show solution
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer:
B
— Not continuous at x = 0
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Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
A.
3 = 2
B.
1 = 2
C.
2 = 1
D.
2 = 4
Show solution
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer:
A
— 3 = 2
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Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Show solution
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer:
A
— 3
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Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer:
A
— Yes
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Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
A.
x = -1
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer:
B
— x = 0
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Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
A.
Continuous everywhere
B.
Continuous at x = 1
C.
Not continuous at x = 1
D.
Continuous for x < 1
Show solution
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer:
C
— Not continuous at x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
A.
x = 0
B.
x = 1
C.
x = 2
D.
x = -1
Show solution
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer:
B
— x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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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
Show solution
Solution
For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.
Correct Answer:
B
— f'(1) = 1
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Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
A.
f(2) = 4
B.
lim x->2 f(x) = 4
C.
Both a and b
D.
None of the above
Show solution
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer:
C
— Both a and b
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Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Show solution
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer:
C
— 4
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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
Show solution
Solution
f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.
Correct Answer:
B
— No
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Q. The maximum value of the function f(x) = -x^2 + 4x + 1 is at x = ?
Show solution
Solution
To find the maximum, we calculate f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. Since f''(x) = -2 < 0, this is a maximum point.
Correct Answer:
B
— 2
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Q. The maximum value of the function f(x) = -x^2 + 4x + 1 is:
Show solution
Solution
The vertex form of a parabola gives the maximum value at x = -b/(2a) = 2. Evaluating f(2) = -2^2 + 4*2 + 1 = 9.
Correct Answer:
A
— 5
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Q. The maximum value of the function f(x) = -x^2 + 4x + 1 occurs at:
A.
x = 2
B.
x = 4
C.
x = 1
D.
x = 3
Show solution
Solution
The vertex of the parabola given by f(x) = -x^2 + 4x + 1 occurs at x = -b/(2a) = -4/(-2) = 2, which gives the maximum value.
Correct Answer:
A
— x = 2
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Q. The minimum value of the function f(x) = x^4 - 8x^2 + 16 is:
Show solution
Solution
Finding the derivative f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. Evaluating f(0) = 16, f(2) = 0, and f(-2) = 0, the minimum value is 0.
Correct Answer:
A
— 0
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Q. The slope of the tangent to the curve y = sin(x) at x = π/4 is:
A.
1
B.
√2/2
C.
√3/3
D.
√2
Show solution
Solution
The derivative f'(x) = cos(x). At x = π/4, f'(π/4) = cos(π/4) = √2/2.
Correct Answer:
B
— √2/2
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Showing 451 to 480 of 574 (20 Pages)
Calculus MCQ & Objective Questions
Calculus is a vital branch of mathematics that plays a significant role in various school and competitive exams. Mastering calculus concepts not only enhances your problem-solving skills but also boosts your confidence during exams. Practicing MCQs and objective questions is essential for effective exam preparation, as it helps you identify important questions and strengthens your understanding of key topics.
What You Will Practise Here
Limits and Continuity
Differentiation and its Applications
Integration Techniques and Fundamental Theorem of Calculus
Applications of Derivatives in Real Life
Definite and Indefinite Integrals
Area Under Curves and Volume of Solids of Revolution
Common Functions and Their Derivatives
Exam Relevance
Calculus is a crucial topic in the CBSE curriculum and is also featured prominently in State Board exams, NEET, and JEE. Students can expect questions that test their understanding of limits, derivatives, and integrals. Common question patterns include solving problems based on real-life applications, finding maxima and minima, and evaluating integrals. Familiarity with these patterns through practice questions will help you excel in your exams.
Common Mistakes Students Make
Confusing the concepts of limits and continuity.
Misapplying differentiation rules, especially for composite functions.
Overlooking the importance of the constant of integration in indefinite integrals.
Failing to interpret the meaning of derivatives in real-world scenarios.
Neglecting to check the domain of functions when solving problems.
FAQs
Question: What are the key formulas I should remember for calculus? Answer: Important formulas include the power rule, product rule, quotient rule for differentiation, and basic integration formulas like ∫x^n dx = (x^(n+1))/(n+1) + C.
Question: How can I improve my speed in solving calculus MCQs? Answer: Regular practice with timed quizzes and focusing on understanding concepts rather than rote memorization can significantly improve your speed.
Start solving practice MCQs today to test your understanding and solidify your calculus knowledge. Remember, consistent practice is the key to success in your exams!
