Chapter Exam

CONTINUITY AND DIFFERENTIABILITY — Online MCQ Test

Q: A function f(x) is continuous at x = a if which of the following conditions is satisfied?

Answer: A. lim(x→a) f(x) = f(a) and the limit exists — Continuity at x = a requires that the limit exists and equals the function value at that point: lim(x→a) f(x) = f(a).

Q: What is the derivative of f(x) = 5x³ + 2x² - 7 with respect to x?

Answer: A. 15x² + 4x — Using power rule: d/dx(5x³) = 15x², d/dx(2x²) = 4x, and d/dx(-7) = 0. Therefore, f'(x) = 15x² + 4x.

Q: Find the derivative of f(x) = eˣ sin(x) using the product rule.

Answer: A. eˣ(sin(x) + cos(x)) — Using product rule: f'(x) = (eˣ)'·sin(x) + eˣ·(sin(x))' = eˣ sin(x) + eˣ cos(x) = eˣ(sin(x) + cos(x)).

Q: Find the left derivative and right derivative of f(x) = |x| at x = 0. Which of the following is correct?

Answer: A. Left derivative = -1, Right derivative = 1 — For x 0: f(x) = x, so right derivative = d/dx(x)|ₓ₌₀ = 1.

Q: A function f(x) is defined as f(x) = {(x² - 9)/(x - 3), x ≠ 3; k, x = 3}. What value of k makes f continuous at x = 3?

Answer: B. k = 6 — Simplify: (x² - 9)/(x - 3) = (x-3)(x+3)/(x-3) = x + 3 for x ≠ 3. So lim(x→3) f(x) = 3 + 3 = 6. For continuity, k = 6.

MATHS · CLASS 12 SECOND PUC · Karnataka State Board

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CONTINUITY AND DIFFERENTIABILITY — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 12 SECOND PUC MATHS — CONTINUITY AND DIFFERENTIABILITY, with answers and explanations. These are sample questions; the exam has its own separate question set.

A function f(x) is continuous at x = a if which of the following conditions is satisfied?

A. lim(x→a) f(x) = f(a) and the limit exists ✓
B. f(a) is defined
C. lim(x→a⁺) f(x) exists
D. f(x) is differentiable at x = a

Answer: A. lim(x→a) f(x) = f(a) and the limit exists
Continuity at x = a requires that the limit exists and equals the function value at that point: lim(x→a) f(x) = f(a).

What is the derivative of f(x) = 5x³ + 2x² - 7 with respect to x?

A. 15x² + 4x ✓
B. 15x² + 4x - 7
C. 5x² + 2x
D. 3x² + 2x

Answer: A. 15x² + 4x
Using power rule: d/dx(5x³) = 15x², d/dx(2x²) = 4x, and d/dx(-7) = 0. Therefore, f'(x) = 15x² + 4x.

Find the derivative of f(x) = eˣ sin(x) using the product rule.

A. eˣ(sin(x) + cos(x)) ✓
B. eˣ cos(x)
C. eˣ sin(x) + eˣ cos(x)
D. sin(x) + eˣ

Answer: A. eˣ(sin(x) + cos(x))
Using product rule: f'(x) = (eˣ)'·sin(x) + eˣ·(sin(x))' = eˣ sin(x) + eˣ cos(x) = eˣ(sin(x) + cos(x)).

Find the left derivative and right derivative of f(x) = |x| at x = 0. Which of the following is correct?

A. Left derivative = -1, Right derivative = 1 ✓
B. Left derivative = 1, Right derivative = -1
C. Left derivative = 0, Right derivative = 0
D. Left derivative = 1, Right derivative = 1

Answer: A. Left derivative = -1, Right derivative = 1
For x < 0: f(x) = -x, so left derivative = d/dx(-x)|ₓ₌₀ = -1. For x > 0: f(x) = x, so right derivative = d/dx(x)|ₓ₌₀ = 1.

A function f(x) is defined as f(x) = {(x² - 9)/(x - 3), x ≠ 3; k, x = 3}. What value of k makes f continuous at x = 3?