Question

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=\ln x\) (a) \([1,4], \quad\) ( b) \([100,103]\)

Short answer

The average rate of change of the function over the interval [1,4] is \(\frac{\ln 4}{3}\) and over the interval [100,103] is \(\frac{\ln 103 - \ln 100}{3}\).

Step-by-step solution

Apply the formula for the first interval [1,4]

The average rate of change of the function over the interval [1,4] is calculated as \(\frac{f(4) - f(1)}{4 - 1}\). The function \(f(x)=\ln x\) is substituted into this formula. Therefore, it becomes \(\frac{\ln 4 - \ln 1}{4 - 1}\).

Simplify the resulting expression

\(\ln 1 = 0\), so the expression simplifies to \(\frac{\ln 4}{3}\). This is the average rate of change of the function over the interval [1,4].

Apply the formula for the second interval [100,103]

The average rate of change of the function over the interval [100,103] is calculated as \(\frac{f(103) - f(100)}{103 - 100}\). Substituting \(f(x)=\ln x\), it becomes \(\frac{\ln 103 - \ln 100}{103 - 100}\).

Simplify the resulting expression

This expression can't be simplified further. Thus, the average rate of change of the function over the interval [100,103] is \(\frac{\ln 103 - \ln 100}{3}\).