Find the number(s) c referred to in the mean value theorem for integrals for each function, over the interval indicated. f ( x ) = 6 x 2 over [-3,4]

Question

Find the number(s) c referred to in the mean value theorem for integrals for each function, over the interval indicated.  f  (  x  )  =  6      x          2       over [-3,4]

jbacapzh · Accepted Answer

Mean Value Theorem For IntegralsIf f is continuous on [a,b] then thereexists a number c in [a,b] such that      ∫          a              b        f  (  x  )  d  x  =  f  (  c  )  [  b  −  a  ]Finding the integral      ∫          −      3              4        6      x          2        d  x  =  [  2      x          3        d  x      ]          −      3              4            ∫          −      3              4        6      x          2        d  x  =  [  128  −  (  −  54  )  ]      ∫          −      3              4        6      x          2        d  x  =  182Applying Mean Value theorem for integrals  f  (  c  )  [  4  −  (  −  3  )  ]  =      ∫          −      3              4        6      x          2        d  xUsing the integral we found earlier7f(c)=182  f  (  c  )  =      182    7  We want c, not f(c)   setting the function equal to 182/7  6      c          2        =      182    7        c          2        =      182    42    c  =  ±            182      42      after some rationalizing  c  =  ±            39        3  Both are values are within our interval, so both are valid values for c

Markus Petty · Accepted Answer

So first we find the slope between the end intervals(-3,54) and (4,96)(96-54)/(4-(-3)) = 6Now we find derivative of f(x) which is 12xand set it equal to 612x= 6x= 1/2

Find the number(s) c referred to in the mean value theorem for integrals for each function, over the interval indicated. f(x)=6x^2 over [-3,4]

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