Engineering Mathematics – I                                 Dr. V. Lokesha         2011 
           
                                                         
                           Engineering Mathematics – I 
                                                (10 MAT11) 
                                                         
                                        LECTURE NOTES 
                                    (FOR I SEMESTER B E OF VTU) 
                                                         
                                                         
                                  VTU-EDUSAT Programme-15 
                                                                      
                 
                 
                 
                                                               
                                                Dr. V. Lokesha 
                                             Professor and Head 
                                   DEPARTMENT OF MATHEMATICS 
                                ACHARYA INSTITUTE OF TECNOLOGY 
                                       Soldevanahalli, Bangalore – 90 
                                                         
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            Engineering Mathematics – I                                 Dr. V. Lokesha         2011 
           
                                                                  
                                                                                                                             
                                                ENGNEERING MATHEMATICS – I 
                 
                                                  Content 
                 
                                    CHAPTER 
                                UNIT I           DIFFERENTIAL CALCULUS – I 
                                UNIT II          DIFFERENTIAL CALCULUS – II 
                                UNIT III         DIFFERENTIAL CALCULUS – III 
                 
                                                                                                                                                          
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
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                        Engineering Mathematics – I                                 Dr. V. Lokesha                                                                                                        2011 
                      
                                                                                                           UNIT -  I 
                                                                                   DIFFERENTIAL CALCULUS – I 
                                  Introduction: 
                                  The mathematical study of change like motion, growth or decay is calculus. The Rate of change of 
                                  given function is derivative or differential. 
                                            
                                  The concept of derivative is essential in day to day life. Also applicable in Engineering, Science, 
                                  Economics, Medicine etc.  
                                   
                                  Successive Differentiation:  
                                  Let y = f (x) --(1) be a real valued function. 
                                                                                                      dy                           1
                                  The first order derivative of y denoted by        or y’ or y or∆  
                                                                                                      dx                  1     
                                                                                                            d2y                             2
                                  The Second order derivative of y denoted by         or y’’ or  y or ∆  
                                                                                                             dx2                   2 
                                   Similarly differentiating the function (1) n-times, successively, 
                                                                                                              dny
                                           th                                                                               n                 n 
                                  the n    order derivative of y exists denoted by         or y  or y  or ∆  
                                                                                                              dxn                    n
                                                                         nd
                                  The process of finding 2  and higher order derivatives is known as Successive Differentiation.  
                                                                                                                       
                                     th                                                                                                               
                                  n  derivative  of  some  standard  functions: 
                                                            ax
                                    1.       y = e  
                                                                         ax
                                  Sol :         y1 = a e  
                                                                    2     ax
                                                  y2 = a  e                    
                                  Differentiating Successively
                                                                          
                                                                    n     ax
                                                  yn = a  e                    
                                                        n     ax           n     ax
                                   ie.         D [e               ] = a  e  
                                                                                 n    x           x
                                   For, a =1   D [e ] = e  
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            Engineering Mathematics – I                                 Dr. V. Lokesha         2011 
           
                                                                                 
                                                                                                
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