Question

find the average rate of change of each function on the given interval.

1. ( f(x)=\frac{x^{2}+3}{x - 2} ;4,9 )
2. ( f(x)=sqrt{2x - 1} ;5,25 )

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\).

Step2: Calculate \(f(4)\) and \(f(9)\) for \(f(x)=\frac{x^{2}+3}{x - 2}\)

• For \(x = 4\):

\(f(4)=\frac{4^{2}+3}{4 - 2}=\frac{16 + 3}{2}=\frac{19}{2}\)

• For \(x = 9\):

\(f(9)=\frac{9^{2}+3}{9 - 2}=\frac{81+3}{7}=\frac{84}{7} = 12\)

Step3: Compute the average rate of change

\(\frac{f(9)-f(4)}{9 - 4}=\frac{12-\frac{19}{2}}{5}=\frac{\frac{24 - 19}{2}}{5}=\frac{\frac{5}{2}}{5}=\frac{1}{2}\)

Step4: Calculate \(f(5)\) and \(f(25)\) for \(f(x)=\sqrt{2x-1}\)

• For \(x = 5\):

\(f(5)=\sqrt{2\times5-1}=\sqrt{10 - 1}=3\)

• For \(x = 25\):

\(f(25)=\sqrt{2\times25-1}=\sqrt{50 - 1}=7\)

Step5: Compute the average rate of change

\(\frac{f(25)-f(5)}{25 - 5}=\frac{7 - 3}{20}=\frac{4}{20}=\frac{1}{5}\)

Answer:

1. The average rate of change of \(f(x)=\frac{x^{2}+3}{x - 2}\) on \([4,9]\) is \(\frac{1}{2}\)
2. The average rate of change of \(f(x)=\sqrt{2x-1}\) on \([5,25]\) is \(\frac{1}{5}\)