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 ( 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 ( 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 given by (\frac{f(b)-f(a)}{b - a}). Here, (a=-5) and (b = - 2), and (f(x)=x^{3}-4x + 5).

Step2: Calculate (f(-5))

Substitute (x=-5) into (f(x)): [ \begin{align*} f(-5)&=(-5)^{3}-4\times(-5)+5\ &=-125 + 20+5\ &=-100 \end{align*} ]

Step3: Calculate (f(-2))

Substitute (x = - 2) into (f(x)): [ \begin{align*} f(-2)&=(-2)^{3}-4\times(-2)+5\ &=-8+8 + 5\ &=5 \end{align*} ]

Step4: Calculate the average rate of change

Using the formula (\frac{f(b)-f(a)}{b - a}), with (a=-5), (b=-2), (f(a)=-100), and (f(b) = 5): [ \begin{align*} \frac{f(-2)-f(-5)}{-2-(-5)}&=\frac{5-(-100)}{-2 + 5}\ &=\frac{5 + 100}{3}\ &=\frac{105}{3}\ &=35 \end{align*} ]

Answer:

(35)