CLASS 11 MATHS CHAPTER-13 LIMITS AND DERIVATIVES

Exercise 13.1

Evaluate the following limits in Exercises 1 to 22.

Question 1. 

Solution :

Given that,

Substituting x = 3, we get

= 3 + 3 = 6

Question 2. 

Solution :

Given that,

Substituting x = π, we get
=(π – 22 / 7)

Question 3. 

Solution :

Given that,

Substituting r = 1, we get
=  π(1)² = π

Question 4. 

Solution :
given limit,

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2) =

                  = (16 + 3) / 2

                  = 19 / 2

Question5. 

Solution :

Given that,

Substituting x = -1, we get

= [(-1)¹⁰ + (-1)⁵ + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

Question6. 

Solution :
Given limit,

Putting x as 0

= [(0 + 1)⁵ – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

=5

Question 7. 

Solution :

Question 8. 

Solution :

Question 9. 

Solution :
Given that,

= [a (0) + b] / c (0) + 1

= b / 1

= b

Question 10. 

Solution :

Question 11. 

Solution :
Given that,

Substituting x = 1

= [a (1)² + b (1) + c] / [c (1)² + b (1) + a]

= (a + b + c) / (a + b + c)

= 1

Question 12. 

Solution :

Question 13. 

Solution :
Given that 

Question 14. 

Solution :

Question 15. 

Solution :

Question 16. 

Solution :

Question 17. 

Solution :

Question18. 

Solution :

Question19. 

Solution :

Question20. 

Solution :

Question21. 

Solution :

Question22. 

Solution :

= 1 x 2

= 2

Question23.Find and where 

Solution :

Question 24. Find where 

Solution :
Given:

.

Question 25. Evaluate where 

Solution :
Given:

.

Question26. Find Where 

Solution :
Given:

Question 27. Find where 

Solution :
Given:

Question 28. Suppose and if what are possible values of a and b?

Solution :
Given: and 

Question 29. Let a₁, a_2,………a_n be fixed real numbers and define a function f (x) = (x – a₁) (x – a₂) ……. (x – a_n).What is  For some a ≠ a₁, a₂, ……. a_n, compute 

Solution :

Question30. If for what values of (s) does exists?

Solution :

Question 31. If the function f(x) satisfies valuate 

Solution :

Question32. If for what integer m and n does both and exist?

Solution :
Given function is  

Exercise 13.2

Evaluate the following limits in Exercises 1 to 22.

Question1. Find the derivative of x² – 2 at x = 10.

Solution :
Here f(x) = x² – 2

Question2.Find the derivative of 99x at x = 100.

Solution :

Let f (x) = x

Then,

Question3. Find the derivative of x at x=1.

Solution :

Let f (x) = 99x,

From first principle

Question4. Find the derivatives of the following functions from first principle:

(i) x³−27 

(ii) (x−1)(x−2)

(iii)1/x²

( iv )x+1/x−1

Solution :
(i)Let f (x) = x³ – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii)Let f (x) = 1 / x²

From first principle, we get

(iv)Let f (x) = x + 1 / x – 1

From first principle, we get

Question5. For the functionprove that f’ (1) =100 f’ (0).

Solution :
 

Question6. Find the derivative of xⁿ + axⁿ⁻¹ + a²xⁿ⁻² + …+aⁿ⁻¹x + aⁿ for some fixed real number a.

Solution :
 

Question7. For some constants a and b find the derivative of:

(i) (x − a) (x − b)

(ii) (ax² + b)²

(iii) x – a / x – b

Solution :
(i) Given:(x − a) (x − b)

(ii)(ax² + b)²

(iii) x – a / x – b

Question8. Find the derivative offor some constant a.

Solution :
 

Question9. Find the derivative of

(i) 2x−3/4

(ii) (5x³+3x−1)(x−1)

(iii) x−3(5+3x)

(iv) x⁵ (3−6x−9)

(iv) x−4(3−4x−5) 

(vi) 2/x+1 − x² /3x−1

Solution :
(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question10. Find the derivative of cos x from first principle.

Solution :
 ,

Question11. Find the derivative of the following functions:

(i) sin x cos x 

(ii) sec x 

(iii) 5sec x + 4cos x

(iv) cosec x 

(v) 3cot + 5cosec x

(vi) 5sin – 6cos x + 7 

(vii) 2tan x – 7sec x

Solution :
(i) 

(ii) 

(iii) 

(iv)

(v)

(vi) 

(vii)

Miscellaneous Exercise

Question1. Find the derivative of the following functions from first principle:

 (i) −x

( ii) (−x)⁻¹

( iii) sin(x+1) 

(iii)cos(x−π/8)

Solution :
(i)Given:f(x) = -x  then

(ii)Given:

then

(iii)Given:f(x) = sin (x + 1)

then

(iv)Given:

then

Question2. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)

Solution :

f(x) = (x + a)

Accordingly,

By first principle,

Question3. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (px+q)(r/x+s)

Solution :
Given:f(x) =

Question4. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b)(cx+d)²

Solution :
Given:f(x) = (ax+b)(cx+d)²

Question5. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question6. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question7. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question8. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question9. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question10. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question11. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question12. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax+b)ⁿ

Solution :

Question13. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):(ax+b)ⁿ (cx+d)^m 

Solution :

since,

Question14. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):sin (x + a) 

Solution :

Question15. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

Solution :

Question16. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question17. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question18. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question19. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinⁿx 

Solution :

Question20. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question21. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question22. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x⁴(5 sin x – 3 cos x) 

Solution :

Question23. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  (x²+1)cosx

Solution :

Question24. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax² + sinx)(p + qcosx)

Solution :

Question25. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x+cosx) (x−tanx)

Solution :

Question26. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question27. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question28. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :

Question29. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

Solution :
let f(x) = (x + sec x) (x – tan x)

By product rule,

Question30. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Solution :