# How do you find the average rate of change of #y=(2x+1)/(x+2)# over [1,3]?

The average rate of change is

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To find the average rate of change of the function ( y = \frac{{2x + 1}}{{x + 2}} ) over the interval ([1, 3]), you first evaluate the function at the endpoints of the interval to get ( y(1) ) and ( y(3) ). Then, you use the formula for average rate of change, which is ( \frac{{f(b) - f(a)}}{{b - a}} ), where ( f(a) ) and ( f(b) ) are the function values at the endpoints of the interval, and ( a ) and ( b ) are the endpoints of the interval. So, you would calculate:

[ y(1) = \frac{{2(1) + 1}}{{1 + 2}} = \frac{3}{3} = 1 ] [ y(3) = \frac{{2(3) + 1}}{{3 + 2}} = \frac{7}{5} ]

Now, use the formula for average rate of change:

[ \text{Average rate of change} = \frac{{y(3) - y(1)}}{{3 - 1}} ]

[ = \frac{{\frac{7}{5} - 1}}{{3 - 1}} ]

[ = \frac{{\frac{7}{5} - \frac{5}{5}}}{{2}} ]

[ = \frac{{\frac{2}{5}}}{{2}} ]

[ = \frac{1}{5} ]

So, the average rate of change of the function ( y = \frac{{2x + 1}}{{x + 2}} ) over the interval ([1, 3]) is ( \frac{1}{5} ).

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