Question

Consider the given function and the given interval.
f(x) = 3 sqrt x, [0, 16]
(a) Find the average value fave of f on the given interval
(b) Find c such that fave = f
(c). (Round your answer to three decimal places.)

Answer 1

Answer:

  (a)  fave = 8

  (b)  c = 64/9

  (c)   c ≈ 7.111

Step-by-step explanation:

(a) The average value of the function is its integral over the interval, divided by the width of the interval.

  $f_{ave}=\displaystyle\frac{1}{16-0}\int_0^{16} {3x^{\frac{1}{2}}} \, dx=\left.\frac{x^{3/2}}{8}\right|_0^{16}=8$

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(b) We want ...

  f(c) = 8

  3√c = 8 . . . . . use f(c)

  √c = 8/3 . . . . . divide by 3

  c = (8/3)² . . . . square

  c = 64/9

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(c) c ≈ 7.111