Continuity and Differentiability — Class 12 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Continuity and Differentiability" — 7 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Continuity and Differentiability" — 7 important questions with detailed answers fo…
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Key Questions Covered:
- Check whether the function f(x) = |x| is continuous at x = 0.
- Determine whether f(x) = 1/(x-2) is continuous at x = 2.
- Find the derivative of f(x) = x² at x = 3 using first principles.
- + 4 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| Check whether the function f(x) = |x| is continuous at x … | ✓ Solved |
| Determine whether f(x) = 1/(x-2) is continuous at x = 2. | ✓ Solved |
| Find the derivative of f(x) = x² at x = 3 using first pri… | ✓ Solved |
Showing 3 of 7 questions
Q1: Check whether the function f(x) = |x| is continuous at x = 0.
To check continuity at x = 0, we verify: lim[x→0⁻] f(x) = lim[x→0⁺] f(x) = f(0)
Given: f(x) = |x|
Step 1: Find f(0)
f(0) = |0| = 0
Step 2: Find left-hand limit as x → 0⁻
For x < 0: f(x) = |x| = -x
lim[x→0⁻] f(x) = lim[x→0⁻] (-x) = 0
Step 3: Find right-hand limit as x → 0⁺
For x > 0: f(x) =...
Q2: Determine whether f(x) = 1/(x-2) is continuous at x = 2.
To check continuity at x = 2, we need to verify if lim[x→2] f(x) exists and equals f(2).
Given: f(x) = 1/(x-2)
Step 1: Check if f(2) is defined
f(2) = 1/(2-2) = 1/0, which is undefined.
Since f(2) is not defined, the function cannot be continuous at x = 2.
Alternatively, examining the limits:
li...
Q3: Find the derivative of f(x) = x² at x = 3 using first principles.
Given: f(x) = x²
We find f'(3) using the definition of derivative.
f'(x) = lim[h→0] (f(x+h) - f(x))/h
At x = 3:
f'(3) = lim[h→0] (f(3+h) - f(3))/h
= lim[h→0] ((3+h)² - 3²)/h
= lim[h→0] ((9 + 6h + h²) - 9)/h
= lim[h→0] (6h + h²)/h
= lim[h→0] (6 + h)
= 6 + 0
= 6
...
Showing 3 of 7 questions. Visit the full page for complete solutions.