Find f'(t) and f(t).f"(t)=t-cos t, f'(0)=6, f(0)=-6

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Find f&#039;(t) and f(t).f&quot;(t)=t-cos t, f&#039;(0)=6, f(0)=-6

Joel Reese · Accepted Answer

f  &quot;  (  t  )  =  t  −  cos  ⁡  t  ,      f    ′    (  0  )  =  6  ,  f  (  0  )  =  −  6Integrating both sicles, we have      f    ′    (  t  )  =            t      2        2    −  sin  ⁡  t  +      c    1  As f&#039;(0)=6  ⇒  6  =  0  −  sin  ⁡  0  +      c    1    ⇒      c    1    =  6  ⇒      f    ′    (  t  )  =            t      2        2    −  sin  ⁡  t  +  6Again Integrating wrt t, we have  ⇒  f  (  t  )  =      1    2              t      3        3    +  cos  ⁡  t  +  6  t  +      c    2  As f(0)=-6  ⇒  −  6  =  0  +  cos  ⁡  0  +  0  +      c    2    ⇒  −  6  =  1  +      c    2    ⇒      c    2    =  −  7  ⇒  f  (  t  )  =            t      3        6    +  cos  ⁡  t  +  6  t  −  7

Find f'(t) and f(t). f"(t)=t-cos t, f'(0)=6, f(0)=-6

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