Commerce Exam > Commerce Notes > Mathematics (Maths) Class 11 > NCERT Solutions Exercise 12.2: Limits and Derivatives - 2
NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Page 2 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Page 3 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Page 4 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Chapter 12 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? 0 0 0 0 00 0 cos cos lim cos cosh sin sinh cos lim cos 1 cosh sin sinh lim cos (1 cosh sin sinh lim 1 cosh sinh cos lim sin lim 1 cosh cos 0 sin 1 lim 0 h h h h hh h x h x h x x x h xx h xx hh xx hh xx h ? ? ? ? ?? ? ?? ? ?? ?? ? ?? ?? ? ? ? ?? ? ?? ?? ???? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 sinh lim 1 sin '( ) sin h and h x f x x ? ?? ? ?? ?? ?? ? ? ? Question 11: Find the derivative of the following functions: (i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x (iv) cosec x (v) 3cot x + 5cosec x (vi) 5sin x - 6cos x + 7 (vii) 2tan x - 7sec x Solution 11: (i) Let f (x) = sin x cos x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim sin cos sin cos lim 1 lim 2sin( )cos 2sin cos 2 h h h f x h f x fx h x h x h x x h x h x h x x h ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ?? ? ? 0 0 1 lim sin2 sin2 2 1 2 2 2 2 2h 2 lim 2cos .sin 2 2 2 h h x h x h x h x x x h ? ? ? ? ? ?? ?? ? ? ? ? ?? ? ?? ?? 0 1 4 2 2h lim cos .sin 2 2 2 h xh h ? ??? ? ?? ?? Page 5 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Chapter 12 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? 0 0 0 0 00 0 cos cos lim cos cosh sin sinh cos lim cos 1 cosh sin sinh lim cos (1 cosh sin sinh lim 1 cosh sinh cos lim sin lim 1 cosh cos 0 sin 1 lim 0 h h h h hh h x h x h x x x h xx h xx hh xx hh xx h ? ? ? ? ?? ? ?? ? ?? ?? ? ?? ?? ? ? ? ?? ? ?? ?? ???? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 sinh lim 1 sin '( ) sin h and h x f x x ? ?? ? ?? ?? ?? ? ? ? Question 11: Find the derivative of the following functions: (i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x (iv) cosec x (v) 3cot x + 5cosec x (vi) 5sin x - 6cos x + 7 (vii) 2tan x - 7sec x Solution 11: (i) Let f (x) = sin x cos x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim sin cos sin cos lim 1 lim 2sin( )cos 2sin cos 2 h h h f x h f x fx h x h x h x x h x h x h x x h ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ?? ? ? 0 0 1 lim sin2 sin2 2 1 2 2 2 2 2h 2 lim 2cos .sin 2 2 2 h h x h x h x h x x x h ? ? ? ? ? ?? ?? ? ? ? ? ?? ? ?? ?? 0 1 4 2 2h lim cos .sin 2 2 2 h xh h ? ??? ? ?? ?? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? 0 00 1 lim cos 2 sinh 2 sinh limcos 2 .lim cos 2 0 .1 cos2 h hh xh h xh h x x ? ?? ?? ?? ?? ?? ?? ? (ii) Let f (x) = sec x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 0 0 0 ' lim sec sec lim 1 1 1 lim cos( ) cos cos cos 1 lim cos cos 2sin sin 11 22 .lim cos cos( ) 2 2sin 11 2 .lim cos h h h h h h f x h f x fx h x h x h h x h x x x h h x x h x x h x x h x h x h xh xh ? ? ? ? ? ? ?? ? ?? ? ?? ?? ?? ? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?? ?? ?? ? ?? ? ? ?? ? 0 00 sin 2 cos( ) sin 2 2 sin 2 2 11 .lim cos 2 cos( ) 2 sin sin 1 22 .lim .lim cos cos( ) 2 1 sin .1. cos cos sec tan h hh h xh h xh h x h x h h x h h x x h x xx xx ? ?? ?? ?? ? ? ? ? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ? ? (iii) Let f (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,Read More
|
75 videos|238 docs|91 tests
|
FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What is the basic definition of a limit in calculus? |  |
Ans. The basic definition of a limit in calculus is the value that a function approaches as the input value approaches a certain point or infinity.
| 2. How can limits be used to find the derivative of a function? | |
Ans. Limits are used in calculus to find the derivative of a function by taking the limit of the difference quotient as the interval approaches zero.
| 3. What is the significance of derivatives in calculus? | |
Ans. Derivatives in calculus represent the rate of change of a function at a given point, providing information about the slope of the function and its behavior.
| 4. How can one determine if a function is continuous at a certain point using limits and derivatives? | |
Ans. A function is continuous at a certain point if the limit of the function as it approaches that point is equal to the function value at that point, and the derivative exists at that point.
| 5. Can limits and derivatives be used to solve real-world problems? | |
Ans. Yes, limits and derivatives are frequently used in solving real-world problems involving rates of change, optimization, and motion. These concepts are essential in various fields such as physics, engineering, and economics.
Related Exams
About this Document
|
1.2K Views |
|
4.87/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 for Commerce 2024 is part of Mathematics (Maths) Class 11 preparation.
The notes and questions for NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 have been prepared according to the Commerce exam syllabus. Information about NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 covers topics
like and NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Example, for Commerce 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for NCERT Solutions Exercise 12.2: Limits and Derivatives - 2.
Introduction of NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 in English is available as part of our Mathematics (Maths) Class 11
for Commerce & NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 in Hindi for Mathematics (Maths) Class 11 course.
Download more important topics related with notes, lectures and mock test series for Commerce
Exam by signing up for free. Commerce: NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
Description
Full syllabus notes, lecture & questions for NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives - Commerce | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 11 | Best notes, free PDF download
Information about NCERT Solutions Exercise 12.2: Limits and Derivatives - 2
In this doc you can find the meaning of NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 defined & explained in the simplest way possible. Besides explaining types of
NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 theory, EduRev gives you an ample number of questions to practice NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 tests, examples and also practice Commerce
tests
|
Explore Courses for Commerce exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
study material
,Free
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,ppt
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,practice quizzes
,Important questions
,Objective type Questions
,Summary
,MCQs
,mock tests for examination
,shortcuts and tricks
,Extra Questions
,Sample Paper
,Previous Year Questions with Solutions
,Exam
,Semester Notes
,Viva Questions
,video lectures
,past year papers
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
;
Additional Information about NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 for Commerce Preparation
NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Free PDF Download
The NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 is an invaluable resource that delves deep into the core of the Commerce exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 now and kickstart your journey towards success in the Commerce exam.
Importance of NCERT Solutions Exercise 12.2: Limits and Derivatives - 2
The importance of NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 cannot be overstated, especially for Commerce aspirants.
This document holds the key to success in the Commerce exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Notes
NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to NCERT Solutions Exercise 12.2: Limits and Derivatives - 2.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Notes on EduRev are your ultimate resource for success.
NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Commerce Questions
The "NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 Commerce Questions" guide is a valuable resource for all aspiring students preparing for the
Commerce exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 on the App
Students of Commerce can study NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the NCERT Solutions Exercise 12.2: Limits and Derivatives - 2,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of NCERT Solutions Exercise 12.2: Limits and Derivatives - 2 is prepared as per the latest Commerce syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup