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CH -5. Continuity and Differentiability


                                                     Work Sheets
                      Answer the following :

                   1.  The function f ( x )   = [ x ] where [ x ] denotes the greatest integer function of x ,is continuous at
                   ( a ) 4                              ( b ) 0                          ( c ) 1.5                       ( d ) none of these
                                                                 5   − 4,              0 <    ≤ 1
                   2.  The value of b for which the function f(x) =                          is continuous at every
                                                                  2
                                                                 4   + 3     ,     1 <    < 2
                       point of  its domain is
                       ( a )   – 1                          ( b )   0                              ( c )     13/3                            ( d )       1
                                                      
                   3.  If y  =   sin⁡    +    )  ,   then   is equal to :
                                  (
                                                      
                       ( a )                                ( b )                          ( c )                            ( d )            
                           2  −1             1−2             2  −1            1−2  
                                                                             
                   4.  The value of c in Role’s theorem for the function f ( x ) =    sinx ,  x ∈ [ 0,    ] is:
                                                                                 3  
                                                                     
                                               
                             
                       ( a )                          ( b )                                ( c )                  ( d )
                           6                 4                     2             4
                                                                      
                                                   – 1
                                 – 1
                   5.  If U  = sin (   2     ) , V =  tan (   2     ) , then    is:
                                   1 +     2          1 −     2                          2
                                                                    1
                       ( a ) 1                        ( b ) x                                ( c )                       ( d )  1 −    
                                                                    2                1 +     2


                                                      5
                   6.  Find the derivative of  log[log(log   )] with respect to x.
                                    3x 4 1 x 2       dy
                                           
                                     
                   7.  If  y cos   1           , find   .
                                       5           dx
                               
                   8.  If  x  e x y , show that   dy    logx
                          y
                                            dx   {log(xe)} 2
                                      1 
                                                         
                                                          1
                   9.  Differentiate x sin x  with respect to sin x .
                                                                    2
                                
                                   
                                1
                                          
                                      
                   10. If  y e  asin x , 1 x 1 then show that  (1 x )  2  d y    x dy    a y 0
                                                                                  
                                                                               2
                                                                  dx 2    dx
                   11. If x a cost logtan        t     , y sint , find   dy  .
                                                 
                                           2               dx
                                                                   2
                                                                               
                                                                                 2
                                                               2
                   12. If x = sint , y =  sinpt, then prove that : (1 −    )             –   x   +    y=  0.
                                                                       2       
                                                              2
                                                         3
                   13. It is given that for the function f(x) = x – 6 x  +  px  +  q on [1 , 3 ], Rolle’s Theorem holds with
                              1
                       c = 2 +  . Find the values of p and q.
                             √3
                   14. Verify Mean Value Theorem for the function f(x)  = 2sinx  + sin2x on [ 0, π ].
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