find the average rate of change of ( k(x) = -x^{3}+8 ) over the interval ( 0,3 ).\nwrite your answer as an…

find the average rate of change of ( k(x) = -x^{3}+8 ) over the interval ( 0,3 ).\nwrite your answer as an integer, fraction, or decimal rounded to the nearest tenth.\nsimplify any fractions.

find the average rate of change of ( k(x) = -x^{3}+8 ) over the interval ( 0,3 ).\nwrite your answer as an integer, fraction, or decimal rounded to the nearest tenth.\nsimplify any fractions.

Answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 0), (b=3), and (f(x)=-x^{3}+8).

Step2: Calculate (f(0)) and (f(3))

  • For (x = 0): (f(0)=-(0)^{3}+8=8).
  • For (x = 3): (f(3)=-(3)^{3}+8=-27 + 8=-19).

Step3: Substitute into the formula

(\frac{f(3)-f(0)}{3 - 0}=\frac{-19-8}{3}=\frac{-27}{3}=-9)

Answer:

(-9)