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 ( 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 ( 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})

Step2: Calculate (f(-1)) and (f(4))

For (f(x)=x^{3}-4x + 5):

  • When (x=-1), (f(-1)=(-1)^{3}-4\times(-1)+5=-1 + 4+5=8)
  • When (x = 4), (f(4)=4^{3}-4\times4+5=64-16 + 5=53)

Step3: Apply the average - rate - of - change formula

Here (a=-1), (b = 4), so (\frac{f(4)-f(-1)}{4-(-1)}=\frac{53 - 8}{4 + 1}=\frac{45}{5}=9)

Answer:

(9)