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Question-1

Find the indicated limit:

Solution:
= by direct substitution
                         = =

Question-2

Find the indicated limit:

Solution:
= = = 0

Question-3

Find the indicated limit:  

Solution:
= = = = 2x

Question-4

Find the indicated limit:

Solution:
= = m(1)m – 1 = m

Question-5

Find the indicated limit:

Solution:
= ×
                        
                         =

                         =

                         =

                         = 2

                     =

Question-6

Find the indicated limit:

Solution:


= ×

= ×

= =

Question-7

Find the indicated limit:

Solution:
= = (nan – 1 formula)

Question-8

Find the indicated limit:

Solution:
=

                    = =

                   = = =

Question-9

Find the indicated limit:

Solution:
= × =
                         =

                         = =

Question-10

Find the indicated limit:

Solution:
= = 1 × =

Question-11

Find the indicated limit:

Solution:
= = 2 cos a = 2 cos

Question-12

Find the indicated limit:

Solution:
= =

Question-13

Find the indicated limit:

Solution:
= = e. (1)5 = e

Question-14

Evaluate the left and right limits of f(x) = at x = 3. Does the limit of f(x) = x 3 exist? Justify your answer.

Solution:

Let x = 3 + h

Then = = = 27



Let x = 3 – h

Then = = 27

Also, = = 9 + 9 + 9 = 27

Question-15

Find the positive integer n such that = 108.

Solution:
= 108

n 3n – 1 = 108

Put n = 4, then 4.33 = 4 × 27 = 108

n = 4

Question-16

Evaluate [Hint: Take ex or esinx as common factor in numerator]

Solution:
=

                      = =

                      = ex

                      = ex

                      = ex [ 1 – (x - sin x) …….] = e0 = 1

Question-17

If f(x) = , f(x) = 1 and f(x) = 1

Solution:
  f(x) = = b/-1 = -b = 1
b = -1
    f(x) = = a = 1
a = 1

f(x) = = 1

f(2) = 1 and f(-2) = 1

Question-18

 Evaluate and . What can you say about ?

Solution:
Let f(x) = where = x            x 0
                                  = -x           x < 0

Then, f(x) = = = 1 

         f(x) = = = -1
 

     f(x) f(x)

does not exist.

Question-19

Compute Hence evaluate .

Solution:
= = = log a – log b = log
= log

Question-20

Without using the series expansion of log(1 + x), prove that = 1

Solution:

Let y = log(1 + x) Then as x 0 , y 0
= = = = 1

Question-21

Differentiate the following with respect to x:
(i) x7 + ex

(ii) log7x + 200
(iii) 3 sinx + 4 cos x – ex
(iv) ex + 3 tanx + log x6
(v) sin 5 + log10x + 2 sec x
(vi) x-3/2 + 8e + 7tanx
(vii)
(viii)

Solution:
(i) y = x7 + ex
       = 7x6 +ex

(ii) y = log7x + 200
        = logex . log10e + 200
     = log10e

(iii) y = 3 sin x + 4 cos x – ex
     = 3 cos x - 4 sin x – ex

(iv) y = ex + 3 tanx + 6 log x
      = ex + 3 sec2x +

(v) y = sin 5 + log10x + 2 sec x
       = sin 5 + logex log10e + 2 sec x
    = 0 +

(vi) y = x-3/2 + 8e + 7tanx
    = -

(vii) y = = x3 + 3x + = x3 + 3x + 3x-1 + x-3
     = 3x2 + 3 – 3x-2 - 3x-4

(viii) =
                        
y = 2x2 – 6x – 4 +
                   = 4x – 6 -

Question-22

Differentiate the following function using quotient rule.

Solution:
Let y =

= = -

Question-23

Differentiate the following function using quotient rule.

Solution:
Let y =

= =

Question-24

Differentiate the following function using quotient rule.

Solution:
Let y =

       =

           =

           =
           =

Question-25

Differentiate the following function using quotient rule.

Solution:
Let y = =
= = = =
e-x

Question-26

Differentiate the following function with respect to x.
log(sinx)

Solution:
y = log(sinx)

Let u = sinx

   = cosx

     y = log u

   = =

   = × = .cos x = cos x = cot x

Question-27

Differentiate the following function with respect to x.
esinx

Solution:
y = esinx

Put u = sinx

= cosx

y = eu

= eu = esinx

= . = esinx. cosx

Question-28

Differentiate the following function with respect to x.

Solution:
y =

Put u = 1 + cotx

= -cosec2 x

y = u1/2

= =

= =

Question-29

Differentiate the following function with respect to x.
tan(logx)

Solution:
y = tan(logx)

Put u = logx

=

y = tan u

= sec2 u

= . = sec2 u. =

Question-30

Differentiate the following function with respect to x.

Solution:
y =

=

     =

     =

Question-31

Differentiate the following function with respect to x.
log sec

Solution:
y = log sec
Put u =
=

  y = log sec u

  y = log v

Put v = sec u

   = sec u tan u

=

= . .

        = . sec tan .
        =
        =

Question-32

Differentiate the following function with respect to x.
log sin(ex + 4x + 5)

Solution:
y = log sin(ex + 4x + 5)

  =

      = (ex + 4)

      = (ex + 4)cot (ex + 4x + 5)

Question-33

Differentiate the following function with respect to x.
sin(x3/2)

Solution:
y = sin(x3/2)

Put u = x3/2

      y = sin u

    =

    = cos u
    = . = cos u. x1/2 = cos x3/2

Question-34

Differentiate the following function with respect to x.
cos()

Solution:
y = cos u

Put u =

= -sin u

=

= . = - sin u . =

Question-35

Differentiate the following function with respect to x.
esin(log x)

Solution:
y = esin(log x)

Put u = logx

=

 y = esinu

Put v = sin u

= cos u = cos(logx)

Put y = ev

= ev
Hence = . = ev . cos(logx). = esin(logx) cos(logs) .

Question-36

Find the indicated limit:

Solution:
= by direct substitution

                          = = 4

Question-37

Find the indicated limit:

Solution:
= = = 0

Question-38

Find the indicated limit:

Solution:
= = = = 2x

Question-39

Find the indicated limit:

Solution:
= = m(1)m – 1 = m

Question-40

Find the indicated limit:

Solution:
= × =

                                                                                              =

                                                                                              =

                                                                                              = 2

                                                                              =

Question-41

Find the indicated limit:

Solution:



= ×

= ×

= =

Question-42

Find the indicated limit:

Solution:
= =    (nan – 1 formula)

Question-43

Find the indicated limit:

Solution:
=

                     = =

                     = = =

Question-44

Find the indicated limit:

Solution:
= × =
                                                                  =

                                                                  = =

Question-45

Find the indicated limit:

Solution:
= = 1 × =

Question-46

Find the indicated limit:

Solution:
= = 2 cos a = 2 cos a

Question-47

Find the indicated limit:

Solution:
= =

Question-48

Find the indicated limit:

Solution:
= = e. (1)5 = e

Question-49

Evaluate the left and right limits of f(x) = at x = 3. Does the limit of f(x) = x 3 exist? Justify your answer.

Solution:
 

Let x = 3 + h

Then = = = 27



Let x = 3 – h

Then = = 27

Also, = = 9 + 9 + 9 = 27

Question-50

Find the positive integer n such that = 108.

Solution:
  = 108

n 3n – 1 = 108

Put n = 4, then 4.33 = 4 × 27 = 108

n = 4

Question-51

Evaluate [Hint: Take ex or esinx as common factor in numerator]

Solution:
=

                      = =

                      = ex

                      = ex

                     = ex [ 1 – (x - sin x) …….] = e0 = 1

Question-52

If f(x) = , f(x) = 1 and f(x) = 1

Solution:
f(x) = = b/-1 = -b = 1
b = -1
   f(x) = = a = 1
a = 1

f(x) = = 1

f(2) = 1 and f(-2) = 1

Question-53

Evaluate and . What can you say about ?

Solution:
Let f(x) = where = x            x 0
                                  = -x           x < 0

Then,

f(x) = = = 1 

f(x) = = = -1
f(x) f(x)

does not exist.

Question-54

Compute Hence evaluate .

Solution:
= = = log a – log b = log

= log

Question-55

Without using the series expansion of log(1 + x), prove that = 1

Solution:
 

Let y = log(1 + x) Then as x 0 , y 0

  =  = = 1

Question-56

Differentiate the following with respect to x:
(i) x7 + ex

(ii) log7x + 200
(iii) 3 sinx + 4 cos x – ex
(iv) ex + 3 tanx + log x6
(v) sin 5 + log10x + 2 sec x
(vi) x-3/2 + 8e + 7tanx
(vii)
(viii)

Solution:
(i) y = x7 + ex
 

       = 7x6 +ex

(ii) y = log7x + 200 = logex . log10e + 200

    = log10e

(iii) y = 3 sin x + 4 cos x – ex

     = 3 cos x - 4 sin x – ex

(iv) y = ex + 3 tanx + 6 log x

     = ex + 3 sec2x +

(v) y = sin 5 + log10x + 2 sec x

        = sin 5 + logex log10e + 2 sec x

    = 0 +

(vi) y = x-3/2 + 8e + 7tanx

     = -

(vii) y = = x3 + 3x + = x3 + 3x + 3x-1 + x-3

    = 3x2 + 3 – 3x-2 - 3x-4

(viii) =

    
y = 2x2 – 6x – 4 +

  = 4x – 6 -

Question-57

Differentiate the following functions with respect to x.
(i) ex cos x
(ii)
(iii) 6 sin x log10 x + e
(iv) (x4 – 6x3 + 7x2 + 4x + 2) (x3 – 1)
(v) (a – b six (1 – 2 cos x)
(vi) cosec x . cotx
(vii) sin2 x
(viii) cos2 x
(ix) (3x2 + 1)2
(x) (4x2 – 1) (2x + 3)
(xi) (3 sec x – 4 cosec x) (2 sin x + 5 cos x)
(xii) x2 ex sin x
(xiii) ex cos x

Solution:
(i) y = ex cos x

= - ex sin x + cos x ex

(ii) y = x1/n log() = x1/n log x
= =
(iii) y = 6 sin x log10 x + e = 6 sin x logex . log10 e + e

= 6 log10 e

(iv) y = (x4 – 6x3 + 7x2 + 4x + 2)(x3 – 1)

= (x4 – 6x3 + 7x2 + 4x + 2) (3x2) + (x3 – 1) (4x3 – 18x2 + 14x + 4)

(v) y = (a – b six (1 – 2 cos x)

  = (a – b sinx) (2 sin x) + (1 – 2 cos x) (-b cosx)

  = 2a sinx – 2b sin2 x – b cos x + 2b cos2x

(vi) y = cosec x. cot x

= -cosec x cosec2x(-cosec x cotx) = -cosec3 x – cot2x cosec x

(vii) y = sin2x = sin x. sin x

= sinx cosx + cosx sinx = 2 sin x cos x = sin 2x

(viii) y = cos2x = cos x cos x

= - cos x sin x - cos x sin x = -2 sin x cos x = -sin 2x

(ix) y = (3x2 +1) (3x2 +1)

= (3x2 +1) 6x + (3x2 +1) 6x = 12x (3x2 +1)

(x) y = (4x2 – 1) (2x + 3)

= (4x2 – 1) (2x + 3)
= (4x2 – 1) (2) + (2x + 3) 8x = 8x2 – 2 + 16x2 + 24x

     = 24x2 + 24x - 2 = 2(12x2 + 12 x – 1)


(xi) y = (3 sec x – 4 cosec x) (2 sin x + 5 cos x)

= (3 sec x – 4 cosec x) (2 cos x – 5 sin x) + (2 sin x tan x + 4 cosec x cot x)

     = 6 sec x cos x - 8

     

     = 6 – 8 cot x - 15 tan x + 20 + 6
     = 26 + 6 = 26 + 6 tan + 20 cot2x

(xii) y = x2 ex sinx

    = x2 ex cos x + ex sinx(2x) + x2 sin x ex

(xiii) y = ex cos x

     = ex . + logx ex + ex log x

Question-58

Differentiate the following function using quotient rule.

Solution:
Let y =
        = = = =

Question-59

Differentiate the following function using quotient rule.

Solution:
Let y =

=

      =

      =

      =

Question-60

Differentiate the following function using quotient rule.

Solution:
 Let y =
          = =

Question-61

Differentiate the following function using quotient rule.

Solution:
Let y =

=


=

Question-62

Differentiate the following function using quotient rule.

Solution:
Let y =

        = =

Question-63

Differentiate the following function using quotient rule.

Solution:
Let y =
=

Question-64

Differentiate the following function using quotient rule.

Solution:
Let y =

        = = -

Question-65

Differentiate the following function using quotient rule.

Solution:
Let y =

        = =

Question-66

Differentiate the following function using quotient rule.

Solution:
Let y =

       =

           =

           =
           =

Question-67

Differentiate the following function using quotient rule.

Solution:
Let y = =

        = = = =
e-x

Question-68

Differentiate the following function with respect to x.

Solution:
 y = log(sinx)

Let u = sinx

= cosx

y = log u

= =

= × = .cos x = cos x = cot x

Question-69

Differentiate the following function with respect to x.
esinx

Solution:
y = esinx

Put u = sinx

= cosx

y = eu

= eu = esinx

= . = esinx. cosx

Question-70

Differentiate the following function with respect to x.

Solution:
y =

Put u = 1 + cotx

= -cosec2 x

y = u1/2

  = 

= =

Question-71

Differentiate the following function with respect to x.
tan(logx)

Solution:
y = tan(logx)

Put u = logx

=

y = tan u

= sec2 u

= . = sec2 u. =

Question-72

Differentiate the following function with respect to x.

Solution:
y =

=

     =

     =

Question-73

Differentiate the following function with respect to x.
log sec

Solution:
y = log sec
Put u =

=

y = log sec u

y = log v

Put v = sec u

= sec u tan u

=

= . .

        = . sec tan .
        =
        =

Question-74

Differentiate the following function with respect to x.
log sin(ex + 4x + 5)

Solution:
y = log sin(ex + 4x + 5)

=

     = (ex + 4)

     = (ex + 4)cot (ex + 4x + 5)

Question-75

Differentiate the following function with respect to x.
sin(x3/2)

Solution:
y = sin(x3/2)

Put u = x3/2

y = sin u

=

= cos u

= . = cos u. x1/2 = cos x3/2

Question-76

Differentiate the following function with respect to x.
cos()

Solution:
y = cos u

Put u =

= -sin u

=

= . = - sin u . =

Question-77

Differentiate the following function with respect to x.
esin(log x)

Solution:
y = esin(log x)

Put u = logx

=

  y = esinu

Put v = sin u

    = cos u = cos(logx)

Put y = ev

    = ev
Hence = . = ev . cos(logx). = esin(logx) cos(logs) .




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