A function is continuous at x = a when lim f(x) equals f(a), and f(x) = |x| is continuous but not differentiable at 0, so this Class 12 sheet separates continuity from differentiability. The key also gives eˣ and 1/x among its derivative answers.
Key Takeaways
Continuity at x = a needs lim f(x) → a = f(a) (Q1).
f(x) = |x| at 0 is continuous but not differentiable (Q2).
The key records eˣ and 1/x among its answers.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.A function f is continuous at x = a if:
(a) lim (x → a) f(x) = f(a)(b) f(a) exists(c) f is one-one(d) lim (x → a) f(x) exists
2.The function f(x) = |x| at x = 0 is:
(a) Discontinuous(b) Continuous but not differentiable(c) Differentiable(d) Not defined
3.The derivative of eˣ with respect to x is:
(a) x·eˣ(b) eˣ(c) eˣ⁻¹(d) ln x
4.The derivative of ln x with respect to x is:
(a) 1/x(b) x(c) ln x(d) eˣ
5.The derivative of sin x² with respect to x is:
(a) cos x²(b) 2x·cos x²(c) 2x·sin x²(d) cos 2x
6.Using the product rule, d/dx (x³ ln x) = ?
(a) 3x² ln x(b) x²(3 ln x + 1)(c) x³/x(d) 3x²
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the condition for a function f to be differentiable at x = a.
8.Is f(x) = |x| differentiable at x = 0? Give a reason.
9.If y = x⁵ + 3x² - 7, find dy/dx.
10.If y = tan x, find dy/dx.
11.If y = e²ˣ, find dy/dx.
12.If y = sin(3x + 1), find dy/dx.
13.State the chain rule for differentiating the composite function f(g(x)).
14.If y = 2x³, find the value of dy/dx at x = 1.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.The position of a particle is s(t) = t³ - 6t² + 9t. Find the velocity of the particle at t = 1 second.
16.A particle moves so that its position is s(t) = 2t² - 3t metres. Find the velocity at t = 2 seconds.
17.The area of a circle of radius r is A = πr². Find dA/dr when r = 4.
18.If y = x² + 2x, find the value of dy/dx at x = 3.
19.The volume of a spherical balloon is V = (4/3)πr³. Find dV/dr when r = 2.
20.The temperature of a body at time t is T(t) = 3t² + 4t + 10. Find the rate of change of temperature at t = 2.
Answer Key
1. a) lim (x → a) f(x) = f(a)
2. b) Continuous but not differentiable
3. b) eˣ
4. a) 1/x
5. b) 2x·cos x²
6. b) x²(3 ln x + 1)
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
When is a function continuous at x = a?
When lim (x → a) f(x) = f(a), option (a) in this worksheet (Q1).
Is f(x) = |x| differentiable at x = 0?
No — it is continuous but not differentiable, option (b) in this worksheet (Q2).
What are some derivative answers in the key?
eˣ and 1/x.
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