Finding original functions of second derivatives expressionsI have an antiderivative problem.Given some function: f ″ ( t ) = t + t And the conditions when the t = 1: f ( 1 ) = 1 ; f ′ ( 1 ) = 2How do I go about deriving the original function given that I must do this with antiderivatives? My logic has been to just take the antiderivative of the original f′′(t) and then take the antiderivative of f′(t) but I am wrong in this approach?

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Finding original functions of second derivatives expressionsI have an antiderivative problem.Given some function:      f    ″    (  t  )  =  t  +      t  And the conditions when the   t  =  1:  f  (  1  )  =  1  ;      f    ′    (  1  )  =  2How do I go about deriving the original function given that I must do this with antiderivatives? My logic has been to just take the antiderivative of the original f′′(t) and then take the antiderivative of f′(t) but I am wrong in this approach?

Blaze Frank · Accepted Answer

Step 1      f    ″    (  t  )  =  t  +      t    =  t  +      t                  1        2                  f    ′    (  t  )  =            t      2        2    +      2    3        t                  3        2              +      c    1    f  (  t  )  =            t      3        6    +      4    15        t                  5        2              +      c    1    t  +      c    2        f    ′    (  1  )  =      1    2    +      2    3    +      c    1    =  2Step 2      c    1    =      5    6    f  (  1  )  =      1    6    +      4    15    +      c    1    +      c    2    f  (  1  )  =      1    6    +      4    15    +  1  +      c    2    =  1      c    2    =  −      13    30

Zion Wheeler · Accepted Answer

Step 1When you integrate, you add an arbitrary constant. Since you have to integrate twice to find f′(t) and f(t), at each step substitute in the given values and solve for the constants.Try that. If you are still stuck then keep reading.First we integrate to get f′(t):      f    ′    (  t  )  =            t      2        2    +      2    3    t      t    +  CStep 2And since we have that       f    ′    (  1  )  =  2  ,,      1    2    +      2    3    +  C  =  2and so   C  =      5    6  Just use this process again to find f.

Finding original functions of second derivatives expressions I have an antiderivative problem. Gi

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