find the average rate of change of the function\n$f(x)=\\sqrt{x - 1}$ on the interval $4\\leq x\\leq9$…

find the average rate of change of the function\n$f(x)=\\sqrt{x - 1}$ on the interval $4\\leq x\\leq9$. recall that\nthe coordinates for the start of the interval are $(4,$\n3).\nwhat are the coordinates for the end of the\ninterval?\n$(9,4)$\n$(9,3)$\n$(9,82)$\nwhat is the average rate of change for this\nfunction on the given interval?\n$\\frac{1}{5}$\n$\\frac{1}{5}$\n$\\frac{1}{5}$

find the average rate of change of the function\n$f(x)=\\sqrt{x - 1}$ on the interval $4\\leq x\\leq9$. recall that\nthe coordinates for the start of the interval are $(4,$\n3).\nwhat are the coordinates for the end of the\ninterval?\n$(9,4)$\n$(9,3)$\n$(9,82)$\nwhat is the average rate of change for this\nfunction on the given interval?\n$\\frac{1}{5}$\n$\\frac{1}{5}$\n$\\frac{1}{5}$

Answer

Explanation:

Step1: Find (f(4)) and (f(9))

The formula for the average rate of change of a function (y = f(x)) on the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here (a = 4) and (b=9). Given (f(x)=\sqrt{x - 1}), then (f(4)=\sqrt{4 - 1}=\sqrt{3}) and (f(9)=\sqrt{9 - 1}=\sqrt{8}=2\sqrt{2}).

Step2: Calculate the average rate of change

The average rate of change (\frac{f(9)-f(4)}{9 - 4}=\frac{\sqrt{9 - 1}-\sqrt{4 - 1}}{9 - 4}=\frac{2\sqrt{2}-\sqrt{3}}{5}). Another way (if there was a typo and the function was (f(x)=\sqrt{x}-1)): If (f(x)=\sqrt{x}-1), then (f(4)=\sqrt{4}-1=2 - 1=1) and (f(9)=\sqrt{9}-1=3 - 1 = 2). Using the formula for the average rate of change (\text{Average Rate of Change}=\frac{f(b)-f(a)}{b - a}), with (a = 4), (b = 9). Substitute into the formula: (\frac{f(9)-f(4)}{9 - 4}=\frac{( \sqrt{9}-1)-(\sqrt{4}-1)}{9 - 4}). Simplify the numerator: ((\sqrt{9}-1)-(\sqrt{4}-1)=(3 - 1)-(2 - 1)=2 - 1=1). The denominator (9 - 4 = 5). So the average rate of change is (\frac{1}{5}).

Answer:

(\frac{1}{5})