find the average rate of change of each function on the given interval.\n1. ( f(x)=\frac{x^{2}+3}{x - 2}…
find the average rate of change of each function on the given interval.\n1. ( f(x)=\frac{x^{2}+3}{x - 2} ;4,9 )\n2. ( f(x)=sqrt{2x - 1} ;5,25 )
Answer
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:
- The average rate of change of (f(x)=\frac{x^{2}+3}{x - 2}) on ([4,9]) is (\frac{1}{2})
- The average rate of change of (f(x)=\sqrt{2x-1}) on ([5,25]) is (\frac{1}{5})