find the average rate of change of ( f(x)=x^{3}-4x + 5 ) over the following intervals.\n(a) from -5 to…

find the average rate of change of ( f(x)=x^{3}-4x + 5 ) over the following intervals.\n(a) from -5 to -2\n(b) from -1 to 4\n(c) from 4 to 6\n(a) the average rate of change from -5 to -2 is 35.\n(b) the average rate of change from -1 to 4 is 9.\n(c) the average rate of change from 4 to 6 is \\( \\square \\).

find the average rate of change of ( f(x)=x^{3}-4x + 5 ) over the following intervals.\n(a) from -5 to -2\n(b) from -1 to 4\n(c) from 4 to 6\n(a) the average rate of change from -5 to -2 is 35.\n(b) the average rate of change from -1 to 4 is 9.\n(c) the average rate of change from 4 to 6 is \\( \\square \\).

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 = 4), (b=6), and (f(x)=x^{3}-4x + 5).

Step2: Calculate (f(4)) and (f(6))

  • For (x = 4): (f(4)=4^{3}-4\times4 + 5=64-16 + 5=53)
  • For (x = 6): (f(6)=6^{3}-4\times6+5=216-24 + 5=197)

Step3: Substitute into the average - rate - of - change formula

(\frac{f(6)-f(4)}{6 - 4}=\frac{197 - 53}{2}) First, calculate the numerator: (197-53 = 144). Then divide by (2): (\frac{144}{2}=72)

Answer:

(72)