Question

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

Find all values of x in the interval for which the function equals its average value.

Answer 1

Answer 2

SOLUTION :

Average value (f average)  of the given function f(x) in interval (-2, 2)

    2

=  ∫    f(x) dx  / (2 - (-2))

  - 2

           2

= 1/4  ∫   4 e^x dx

        - 2

                          2

= 1/4 *  4 * [e^x]                          

                        - 2                                                    

= 1 * (e^2 - e^(-2))

= 7.2537 = 7.254  approx. (ANSWER)

 

Value of x for which f(x) = 7.2537 :

f(x) = 4 e^x  = 7.2537

Taking natural log :

=> lln(4 * e^x) = ln(7.2537)

=> ln(4) + x = ln(7.2537)

=> x = ln(7.2537) - ln(4)

=> x = 0.595   (ANSWER).