Let f be continuous on I = [ a , b ] and let H : I → R be defined by H ( x ) = ∫ x b f ( t ) d t , x ∈ I. Find H ′ ( x ).

Question

Let   f be continuous on   I  =  [  a  ,  b  ] and let   H  :  I  →      R   be defined by   H  (  x  )  =      ∫    x    b    f  (  t  )  d  t  ,  x  ∈  I. Find       H    ′    (  x  ).

engaliar0l · Accepted Answer

Notice that  H  (  x  )  =      ∫    a    b    f  (  t  )    d  t  −      ∫    a    x    f  (  t  )    d  tthe first term is constant and thus has   0 derivative.Another approach would be  H  (  x  )  =      ∫    x    b    f  (  t  )    d  t  =  −      ∫    b    x    f  (  t  )    d  t

Willow Pratt · Accepted Answer

Going back to fundamentals, since   H  (  x  )  =      ∫    x    b    f  (  t  )  d  t,                    H        (        x        +        h        )        −        H        (        x        )                    =                  ∫                      x            +            h                    b                f        (        t        )        d        t        −                  ∫          x          b                f        (        t        )        d        t                                  =                  ∫                      x            +            h                    b                f        (        t        )        d        t        −        (                  ∫          x                      x            +            h                          f        (        t        )        d        t        +                  ∫                      x            +            h                    b                f        (        t        )        d        t        )                                  =        −                  ∫          x                      x            +            h                          f        (        t        )        d        t                (this is where the &quot;-&quot; comes from)                            so                                                  H            (            x            +            h            )            −            H            (            x            )                    h                            =        −                  1          h                          ∫          x                      x            +            h                          f        (        t        )        d        t                                  →        −        f        (        x        )         as         h        →        0

Let f be continuous on I = [ a , b ] and let H : I →

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