# How do you find the average rate of change of #f(x) = 3/(x-2)# over the interval [4,7]?

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So the average rate of change is

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To find the average rate of change of a function ( f(x) ) over an interval ([a, b]), you can use the formula:

[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} ]

In this case, the function is ( f(x) = \frac{3}{x - 2} ) and the interval is ([4, 7]).

Substitute ( a = 4 ) and ( b = 7 ) into the formula and evaluate ( f(4) ) and ( f(7) ):

[ f(4) = \frac{3}{4 - 2} = \frac{3}{2} ] [ f(7) = \frac{3}{7 - 2} = \frac{3}{5} ]

Now, plug these values into the formula:

[ \text{Average rate of change} = \frac{f(7) - f(4)}{7 - 4} = \frac{\frac{3}{5} - \frac{3}{2}}{7 - 4} ]

[ \text{Average rate of change} = \frac{\frac{3}{5} - \frac{15}{10}}{3} = \frac{\frac{3}{5} - \frac{3}{2}}{3} ]

[ \text{Average rate of change} = \frac{\frac{6}{10} - \frac{15}{10}}{3} = \frac{-\frac{9}{10}}{3} ]

[ \text{Average rate of change} = -\frac{9}{10} \times \frac{1}{3} = -\frac{3}{10} ]

So, the average rate of change of ( f(x) = \frac{3}{x - 2} ) over the interval ([4, 7]) is ( -\frac{3}{10} ).

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