Chapter 5. CONTINUITY AND DIFFERENTIABILITY

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.1

1. Prove that the function  is continuous at , at  and at .

Solution:  We have,

At   

Therefore,  is continuous at x = 0 .

At  

Therefore,  is continuous at x =  – 3 .

At

Therefore,  is continuous at x = 5 .

2. Examine the continuity of the function  at .

Solution:  We have,  

At x = 3

Therefore,  is continuous at  .

3. Examine the following functions for continuity.

(a)    (b)     (c)    (d)  

Solution:

4. Prove that the function is continuous at   , where  is a positive integer .

Solution:  We have, 

At  

Therefore,  is continuous at  .

5. Is the function  defined by  continuous at   ? at  ? at  ?

Solution:  Given,

At  

Putting  as when , then

Putting  as when  , then

At  

Putting  as when , then

Therefore,  is not continuous at x = 1 .

At  

 

Therefore,  is continuous at  .

Find all points of discontinuity of  , where  is defined by

6.   

Solution: We have,   

At x = 2

Putting  as when , then

Putting  as when  , then

 

Therefore,  is  discontinuous at x = 2 .

7.

Solution:  Given, the function

At

Putting  as when , then

Putting  as when , then

Therefore,  is continuous at  .

At

Putting  as when , then

Putting  as when , then

Therefore,  is discontinuous at x = 3

8. 

Solution: Given, the function

At  

Putting  as when , then

Putting  as when , then

Therefore,  is discontinuous at x = 0 .

9. 

Solution:  Given, the function

At

Putting  as when , then

Therefore,  is continuous at x = 0 .

10.

Solution:  Given, the function      

At x = 1

Putting  as when , then

Putting  as when , then

Therefore,  is continuous at x = 1 .

11.

Solution: Given, the function

At x = 2

Putting  as when , then 

Putting  as when , then

Therefore,  is continuous at x = 2 .

12.

Solution: Given, the function  

At x = 1

Putting  as when , then 

Putting  as when , then

Therefore,  is discontinuous at x = 1 .

13. Is the function defined by a continuous function ?

Solution:  Given, the function      

At x = 1

Putting  as when , then

Putting  as when , then

Therefore,  is discontinuous at x = 1 .

Discuss the continuity of the function , where is defined by

14.       

Solution: Given, the function      

At x = 1

And 

Therefore, f(x) is not continuous at x = 1

At x= 3

and 

Therefore, f(x) is not continuous at x = 3 .

15.       

Solution:  Given, the function      

At x = 0

Putting  as when , then

Therefore, f(x) is continuous at x  .

At x = 1 

                          

Putting  as when , then

Therefore, f(x) is discontinuous at x = 1.

16.   

Solution:  Given, the function

At  

                                             

Putting  as when , then

Therefore,  is continuous at   .

At  

Putting  as when , then

Therefore,  is continuous at   .

17. Find the relationship between  and  so that the function  defined by

    is continuous at  .

Solution:  Given, the function  

At x = 3

Putting  as when , then

Putting  as when , then

Therefore, f(x) is continuous at x = 3 .

18. For what value of  is the function defined by   continuous at  ? What about continuity ?

Solution:  Given, the function       

At x = 0

 

Putting  as when , then

Putting  as when , then

Therefore,  is not continuous at  for any values of   .

At x = 1                                                       

Putting  as when , then

Putting  as when , then

 Therefore,  is continuous at  for any values of  .

19. Show that the function defined by  is discontinuous at all integral points . Here  denotes the greatest integer less than or equal to  .

Solution:

20. Is the function defined by  continuous at ?

Solution:

21. Discuss the continuity of the following functions :

(a)   (b)    (c)  

Solution:

22. Discuss the continuity of the cosine , cosecant , secant and cotangent functions .

Solution:

23. Find all points of discontinuity of  , where

Solution: Given, the function

At x = 0

Putting  as when , then

Putting  as when , then

Therefore,  is not discontinuous at   .

24. Determine if  defined by  is a continuous function ?

Solution: Given, the function

 At x = 0

 

Putting  as when , then

Putting  as when , then

Therefore,  is continuous at   .

25. Examine the continuity of  , where  is defined by

Solution: Given, the function

At x = 0

Putting  as when , then

Putting  as when , then

Therefore,  is continuous at   .

Find the values of  so that the function  is continuous at the indicated point in Exercises 26 to 29 .

 26.       At

Solution: Given, the function       At

Putting   as when , then

  

Putting as   when , then

  

Therefore,  is continuous   .

Therefore, the value of k is 6 .

27. at x = 2.

Solution : Given, the function

At x = 2

Putting  as when , then

  

 

      

         

Therefore, the value of k is .

28. at  

Solution:  Given, the function

At 

Putting  as when , then

Putting  as when , then

 

        

    

   

Therefore, the value of k is .

29.  at x = 5 .

Solution: Given, the function 

At x= 5

Putting  as when , then 

Putting  as when , then

 

    

         

Therefore, the value of k is .

30. Find the values of  and  such that the function defined by

  is a continuous function .

Solution: Given, the function

At x= 2

Putting  as when , then

At x = 10

Putting  as when , then

Putting  in (i) , we get

Therefore, the value of  and  .

31. Show that the function defined by  is a continuous function .

Solution:

32. Show that the function defined by  is a continuous function .

Solution:

33. Examine that  is a continuous function .

Solution:

34. Find all the points of discontinuity of  defined by  .

Solution:

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.2

Differentiate the functions with respect to  in Exercises 1 to 8 .

1.     2.     3.      4.      5.    6.       7.       8.  

Solution:  We have,    

Let    

Differentiating the function with respect to x , we get

2.   

Solution: We have,  

Let    

Differentiating the function with respect to x , we get

 

 

 

3.  

Solution:   We have,

Let   

Differentiating the function with respect to  , we get

 

4.   

Solution:  We have ,  

 

Differentiating the function with respect to  , we get

 

5.

Solution: We have, 

Let 

Differentiating the function with respect to  , we get

6.  

Solution: We have,

Let

Differentiating the function with respect to  , we get

7.      

Solution : We have,     

Let      

Differentiating the function with respect to  , we get

 

 

 

 

 

8.  

Solution: We have,

Let

Differentiating the function with respect to x  , we get

 

 

 

9. Prove that the function  given by  is not differentiable at .

10. Prove that the greatest integer function defined by  is not differentiable at  and x = 2 .

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.3

Find in the following :

1.

Solution :  We have, 

Differentiating the both sides with respect to x , we get

2.

Solution: We have,

Differentiating the both sides with respect to x , we get

3.

Solution:  We have,  

Differentiating the both sides with respect to x , we get

4.

Solution:  We have,  

Differentiating the both sides with respect to x , we get

5.

Solution: We have,

Differentiating the both sides with respect to x , we get

6.

Solution:  We have,

Differentiating the both sides with respect to x , we get

7.   

Solution:  We have,

Differentiating the both sides with respect to x , we get

8.    

Solution: We have,

Differentiating the both sides with respect to x , we get

9. 

Solution: We have,

Let   and

Differentiating the both sides with respect to x  , we get

10.

Solution: We have,

Let   and

 Differentiating the both sides with respect to x , we get

11.

Solution:  We have, 

Let   and

 Differentiating the both sides with respect to x , we get

12.

Solution: We have,

Let   and

Differentiating the both sides with respect to x , we get

 

13.

Solution: We have,

Let   and

Differentiating the both sides with respect to x , we get

14.

Solution: We have,

Let   and

Differentiating the both sides with respect to x , we get

15.

Solution: We have,

Let  and

Differentiating the both sides with respect to x , we get

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.4

Differentiate the following w.r.t.x  :

1.        2.    3.    4.    5.      6.   7.      8.    9.   10.

1.

Solution:  let

Differentiating w.r.t. x , we get

 

 

 2.   

Solution: Let    

Differentiating w.r.t. x , we get

 

 

3.    

Solution:  Let 

Differentiating w.r.t. x , we get

4.   

Solution:  Let     

Differentiating w.r.t. x , we get

 

5.    

Solution: Let 

Differentiating w.r.t. x , we get

6.

Solution:  let 

Differentiating w.r.t. x , we get

7.    

Solution: Let 

Differentiating w.r.t. x , we get

 

8.   

Solution:  Let

Differentiating w.r.t. x , we get

9.

Solution:  Let

Differentiating w.r.t. x , we get

10.

Solution: Let

Differentiating w.r.t. x , we get

 

 

 

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.5

Differentiate the functions given in Exercises 1 to 11 w.r.t.  .

1.     2.      3.     4.   5.   6.    7.   8.    9.   10.    11.

1.  

Solution:  Let

Differentiating the functions w.r.t.  , we get

 

  

 

 

 

2.

Solution:  Let

3.   

Solution : Let    

 

Differentiating the functions w.r.t. x , we get

 

4.  

Solution:  let

Differentiating the functions w.r.t. x , we get

Where,   and

Now, 

 

Differentiating the functions w.r.t. x , we have

and

  

Differentiating the functions w.r.t. x , we have

 

 

 

 

So,    

5.  

Solution: let

Differentiating the functions w.r.t. x , we have

6.   

Solution:  let   

Differentiating the functions w.r.t. x , we have

Where,  and

Differentiating the functions w.r.t. x , we have

 

 

 

 

 

And 

 

Differentiating the functions w.r.t. x , we have

 

 

7.  

Solution:  let  

Differentiating the functions w.r.t. x , we have

Where,  and

Now, 

Differentiating the functions w.r.t. x , we have

And

Differentiating the functions w.r.t. x , we have

8.   

Solution:  let   

Differentiating the functions w.r.t. x , we have

Where,   and

Now ,

  Differentiating the functions w.r.t. x , we have

and 

Differentiating the functions w.r.t. x , we have

 

 

9.  

Solution : let

Differentiating the functions w.r.t. x , we have

 

Where,   and

Now ,

 

  Differentiating the functions w.r.t. x , we have

and

Differentiating the functions w.r.t. x , we have

 

 

 

 

 

10.

Solution:  let

Where,    and

Now, 

 

Differentiating the functions w.r.t. x , we have

 And 

Differentiating the functions w.r.t. x , we have

11.  

Solution: let  

Differentiating the functions w.r.t. x , we have

 

Where,  and

Now,

 

Differentiating the functions w.r.t. x , we have

and

 

Differentiating the functions w.r.t. x , we have

 

 

 

 

Find of the functions given in Exercises 12 to 15 .

12.   13.   14.   15.  

12.  

Solution:  We have,  

Differentiating the functions w.r.t. x , we have

 

Where,  and

Now,    

Differentiating the functions w.r.t. x , we have

 

 

and 

 

Differentiating the functions w.r.t. x , we have

13.   

Solution:  We have,  

Differentiating the functions w.r.t. x , we have                                                  

14.  

Solution:  We have,

  

Differentiating the functions w.r.t. x , we have

15.  

Solution: We have,  

Differentiating the functions w.r.t. x  , we have

16. Find the derivative of the function given by  and hence find  .

Solution:  Given, the function

 

  

17. Differentiate  in three ways mentioned below :

(i) by using product rule

(ii) By expanding the product to obtain a single polynomial .

(iii) By logarithmic differentiation .

Do they all give the same answer ?

Solution:  (i) Let

Differentiating the functions w.r.t. x  , we have

 

 

 

(ii) let

(iii)  let

Differentiating the functions w.r.t. x  , we have

18. If  and  are functions of  , then show that    in two ways-first by repeated application of product rule , second by logarithmic differentiation .

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.6

If  and  are connected parametrically by the equations given in Exercises 1 to 10 , without eliminating the parameter, Find  .

1.    2.   3.  4.

5.   6.  

7.

8.    9.

10.

11. If  , show that 

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.7

Find the second order derivatives of the functions given in Exercises 1 to 10 .

1.     2.    3.   4.   5.    6.   7.     8.  

9.   10.  

11. If , prove that  .

12. If   ,find in terms of  alone .

13. If  , show that  .

14. If , show that  .

15. If , show that   .

16. If  , Show that  .

17. If  , show that  .

Class 12 Mathematics Chapter 5 Continuity and Differentiability Exercise 5.8

1. Verify Rolle’s theorem for the function ].
2. Examine if Rolle’s theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s theorem from these example?
(i)   for  

(ii)   for
(iii)   for
3. If   is a differentiable function and if  does not vanish anywhere, then prove that
4. Verify Mean Value Theorem, if  in the interval  where  and .
5. Verify Mean Value Theorem, if   in the interval  , where  and  . Find all  for which  .
6. Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

Class 12 Mathematics Chapter 5 Continuity and Differentiability Miscellaneous Exercise on Chapter 5

Differentiate w.t.r.  the function in Exercises 1 to 11 .

1.  

2.  

3.   

4.  

5.     

6.

7.  

8.  , for some constant  and b .

9.  

10.   , for some fixedand x  .

11.  , for .

12. If  , if  .

13. Find , if  ,

14. If  , for,  , prove that   .

15. If , for some  , prove that    is a constant independent of  and .

16. If  , with , prove that    .

17. If x and  , find .

18. If , show that  exists for all real  and find it .

19. Using mathematical induction prove that   for all positive integers  .

20. Using the fact that and the differentiation, obtain the sum formula for cosines .

21. Does there exist a function which is continuously everywhere but not differentiable at exactly two points ? Justify your answer .

22. If y  , prove that  .

23. If  , show that  .

Solution: