Question

Find the average value of the function over the given interval. (Round your answer to three decimal places.) f(x)

Answer 1

Complete Question  

Find the average value of the function over the given interval. (Round your answer to three decimal places.) f(x) = 12e^x, [−6, 6] Find all values of x in the interval for which the function equals its average value. (Enter your answers as a comma-separated list. Round your answers to three decimal places.)

Answer:

The  average is  

  $AV = 403$

The value of  x is

   $x = 4$

Step-by-step explanation:

From the question we are told that

  The equation is  $f(t) = 12e^x$

  The  points consider is  [-6 , 6]

Generally the average value of the function over the given interval is mathematically represented as

          $AV = \frac{1}{z-w} \int\limits^ z_w {f(x)} \, dx$

          $AV = \frac{1}{ 6 - (-6)} \int\limits^{6}_{-6} { 12e^x} \, dx$

          $AV = \frac{1}{12 } e^x| \left 6} \atop {-6}} \right.$

          $AV = e^6 -e^{-6}$

          $AV = 403$

Generally when the function equal the average we have that

        $f(x) = 12e^{x} = 403$

        $e^{x} = 34$

          $x = ln(34)$

        $x = 4$