Question

calculate the average rate of change from x = 5 to x = 10 on the function f(x) = \frac{1}{2x}.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 5\), \(b=10\), and \(f(x)=\frac{1}{2x}\).

Step2: Calculate \(f(5)\) and \(f(10)\)

• For \(x = 5\), \(f(5)=\frac{1}{2\times5}=\frac{1}{10}\).
• For \(x = 10\), \(f(10)=\frac{1}{2\times10}=\frac{1}{20}\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(10)-f(5)}{10 - 5}=\frac{\frac{1}{20}-\frac{1}{10}}{5}\).
First, simplify the numerator: \(\frac{1}{20}-\frac{1}{10}=\frac{1 - 2}{20}=-\frac{1}{20}\).
Then, \(\frac{-\frac{1}{20}}{5}=-\frac{1}{20}\times\frac{1}{5}\).

Answer:

\(-\frac{1}{100}\)