DIFFERENTIAL CALCULUS

DIFFERENTIAL  CALCULUS

     

     Differential  calculus  is  a  sub  field   of  calculus  concerned  with  the  study  at   which  quantities  changes.  The  primary  objects  of  differential  calculus  is  derivative   of  a  function.

         Differential  calculus  and  integral  calculus  are  connected  by   the   fundamental  theorem   of  calculus.

PHYSICS 

                  Calculus  has  play an  important  role  in  physics,  Many  physical  processes  are  described  by  equations  involving   derivatives   called  differential  equation .  PHYSICS is  a  particular  way  of  quantity  changes nd   the   concept  of  " time   derivatives "  .   

                                    FOR  an  example  If  an  object  position  on a  line  is  given  by

                                      x(t) =  -16t2  +  16t  -  32

                                    then  the  object  velocity  is  

                                      x'(t) =  -32t  + 16

                                   and  the  object  accleration  is  

                                 -32  

  

                    

LIMITS

LIMITS

                              A  limits  is  a  value  that  a  function  or  sequences  "approaches" as  the  input  or  index  approaches  same  value  .  Limits  are  essential  to  calculus  and  are  used  to  defined  continuity, derivatives  and  integrals.

           

LIMITS  OF  A  FUNCTION

      Suppose  f  is  a  real  valued  function  and  c  is  a  real  number.  then  the  expression,

             It  means  that  f(x) can  made  to  be  as  close  to  L  by  desired  making  X  sufficiently  close  to  c.  it  means  that  above  equation  can  be  made  as  "the  limits  of  F  of  X,as  approaches  to  c,that  is  L.

LIMITS  OF  A  SEQUENCE  

Considered  the  following  sequences,  1.79,  1.799,  1.7999.........  it  can  be  observed  that  number  are  approaches  to  1.8 ,  the  limit  of  the  sequences.Formally  suppose  that

         a1, a2,  are  the  sequences  of  a  real  number.It  can  be  stated  that  a  real  number is  L of the  limit  of  this  sequence.

       

Which  is  read  as

    "The  limit  of  an  as  n  approaches  infinity  equals L"

INDEFINITE INTEGRAL , DIFFERENTIATION , DEFINITE INTEGRAL

INDEFINITE INTEGRAL 

FORMULAS

DIFFERENTIATION   FORMULAS 

DEFINITE  INTEGRAL  FORMULAS

Trigonometry Formulas

• sin²x + cos²x = 1
• tan²x + sec²x = 1
• cosec²x + cot²x = 1
• sin2x = 2sinxcosx
• sinx = 2sinx|2cosx|2
• cos2x = 2cos²x-1
• cos2x = 1-2sin²x
• cos2x = 1 - tan²x/1 + tan²x
• cos2x = cos²x - sin²x
• sin2x = 2tanx/1 + tan²x
• 1 - cosx = 2cos²x/2
• 1 + cosx = 2sin²x/2
• tan2x = 2tanx/1-tan²x
• sin3x = 3sinx - 4sin³x
• cos3x = 4cos³x - 3cosx
• tan3x = 3tanx - tan³x/1 - 3tan²x
• sin(x+y) = sinxcosy + cosxsiny
• sin(x-y) = sinxcosy - cosxsiny
• cos(x+y) = cosxcosy - sinxsiny
• cos(x-y) = cosxcosy + sinxsiny
• tan(x+y) = tanx + tany/1 - tanxtany
• tan(x-y) = tanx - tany/1 + tanxtany
• cot(x+y) = cotxcoty - 1/cotx + coty
• cot(x-y) = cotxcoty + 1/coty - cotx
• sin(x+y) * sin(x-y) = sin2x - sin2y or cos2y - cos2x
• cos(x+y) * cos(x-y) = cos2x - sin2y or cos2y - sin2x
• sin(x+y) + sin(x-y) = 2sinxcosy
• sin(x+y) - sin(x-y) = 2cosxsiny
• cos(x+y) + cos(x-y) = 2cosxcosy
• cos(x-y) - cos(x+y) = 2sinxsiny
• sinC + sinD = 2sin(C+D)/2*cos(C-D)/2
• sinC - sinD = 2cos(C+D)/2*sin(C-D)/2
• cosC + cosD = 2cos(C+D)/2*cos(C-D)/2
• cosC - cosD = 2sin(C+D)/2*sin(D-C)/2