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.
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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.
- Find the derivative of f(x) = sin(x).
- Check if f(x) = x² is differentiable at x = 1.
- Find the derivative of f(x) = e^x.
- + 1 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 |
| Find the derivative of f(x) = sin(x). | ✓ Solved |
| Check if f(x) = x² is differentiable at x = 1. | ✓ Solved |
| Find the derivative of f(x) = e^x. | ✓ Solved |
Showing 6 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) = |x| = x
lim[x→0⁺] f(x) = lim[x→0⁺] x = 0
Step 4: Check continuity
Since lim[x→0⁻] f(x) = 0 = lim[x→0⁺] f(x) = f(0)
CONCLUSION: f(x) = |x| is continuous at x = 0.
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:
lim[x→2⁻] f(x) = 1/(2⁻ - 2) = 1/0⁻ = -∞
lim[x→2⁺] f(x) = 1/(2⁺ - 2) = 1/0⁺ = +∞
The left and right limits are not equal and both are infinite.
CONCLUSION: f(x) = 1/(x-2) is NOT continuous at x = 2. In...
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
CONCLUSION: f'(3) = 6
Q4: Find the derivative of f(x) = sin(x).
Given: f(x) = sin(x)
We find f'(x) using the definition of derivative.
f'(x) = lim[h→0] (f(x+h) - f(x))/h
= lim[h→0] (sin(x+h) - sin(x))/h
Using the formula: sin(A) - sin(B) = 2 cos((A+B)/2) sin((A-B)/2)
sin(x+h) - sin(x) = 2 cos((x+h+x)/2) sin((x+h-x)/2)
= 2 cos((2x+h)/2) sin(h/2)
= 2 cos(x + h/2) sin(h/2)
f'(x) = lim[h→0] (2 cos(x + h/2) sin(h/2))/h
= lim[h→0] cos(x + h/2) × (2sin(h/2)/h)
= lim[h→0] cos(x + h/2) × (sin(h/2)/(h/2))
...
Q5: Check if f(x) = x² is differentiable at x = 1.
To check if f(x) = x² is differentiable at x = 1, we check if the left and right derivatives are equal.
Left derivative at x = 1:
f'(1⁻) = lim[h→0⁻] (f(1+h) - f(1))/h
= lim[h→0⁻] ((1+h)² - 1)/h
= lim[h→0⁻] (1 + 2h + h² - 1)/h
= lim[h→0⁻] (2h + h²)/h
= lim[h→0⁻] (2 + h)
= 2
Right derivative at x = 1:
f'(1⁺) = lim[h→0⁺] (f(1+h) - f(1))/h
= lim[h→0⁺] ((1+h)² - 1)/h
= lim[h→0⁺] (2h + h²)/h
= lim[h→0⁺] (2 + h)
= 2
Since f'(1⁻) = f'(1⁺)...
Q6: Find the derivative of f(x) = e^x.
Given: f(x) = e^x
We find f'(x) using the definition of derivative.
f'(x) = lim[h→0] (f(x+h) - f(x))/h
= lim[h→0] (e^(x+h) - e^x)/h
= lim[h→0] (e^x × e^h - e^x)/h
= lim[h→0] e^x(e^h - 1)/h
= e^x × lim[h→0] (e^h - 1)/h
We know that lim[h→0] (e^h - 1)/h = 1 (a standard limit)
f'(x) = e^x × 1 = e^x
CONCLUSION: d/dx[e^x] = e^x
This shows that the exponential function is its own derivative.
Showing 6 of 7 questions. Visit the full page for complete solutions.
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