Question

1. let ( f(x)=(x - 7)^0 ). find an interval of ( x ) on which the average rate of change of ( f ) is 0.

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Given \(f(x)=(x - 7)^{0}\), and by the zero - exponent rule \(a^{0}=1\) for \(a
eq0\). So \(f(x)=1\) for \(x
eq7\).

Step2: Calculate \(f(b)-f(a)\)

Let \(a\) and \(b\) be two values in the domain of \(f(x)\) (where \(a
eq7\) and \(b
eq7\)). Then \(f(a) = 1\) and \(f(b)=1\). So \(f(b)-f(a)=1 - 1=0\).

Step3: Determine the interval

Since \(\frac{f(b)-f(a)}{b - a}=\frac{0}{b - a}=0\) (for \(b
eq a\) and \(a
eq7\), \(b
eq7\)). An example of an interval is \([1,2]\) (any interval \([a,b]\) where \(a
eq7\), \(b
eq7\) will work).

Answer:

An interval such as \([1,2]\) (any interval \([a,b]\) with \(a
eq7\) and \(b
eq7\) is valid).