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

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To find the average rate of change of ( y = -\frac{1}{x+2} ) over the interval ([-1, -\frac{1}{2}]), you use the formula:

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

Where: ( f(a) ) represents the value of the function at the lower bound of the interval, ( f(b) ) represents the value of the function at the upper bound of the interval, ( a ) is the lower bound of the interval, and ( b ) is the upper bound of the interval.

Plugging in the values:

[ f(-1) = -\frac{1}{-1+2} = -\frac{1}{1} = -1 ] [ f\left(-\frac{1}{2}\right) = -\frac{1}{\left(-\frac{1}{2}\right)+2} = -\frac{1}{\frac{1}{2}+2} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} ]

[ \text{Average Rate of Change} = \frac{-\frac{2}{5} - (-1)}{-\frac{1}{2} - (-1)} = \frac{-\frac{2}{5} + 1}{\frac{1}{2}} = \frac{\frac{3}{5}}{\frac{1}{2}} = \frac{3}{5} \times 2 = \frac{6}{5} ]

So, the average rate of change of ( y = -\frac{1}{x+2} ) over ([-1, -\frac{1}{2}]) is ( \frac{6}{5} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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