Question

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

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:

\(9\)