# Consider the function #f(x) = (x)/(x+2)# on the interval [1, 4], how do you find the average or mean slope of the function on this interval?

The mean slope is

Here,

Therefore,

So,

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To find the average or mean slope of the function ( f(x) = \frac{x}{x + 2} ) on the interval ([1, 4]), you can use the formula for the average rate of change or slope.

The average slope of a function over an interval ([a, b]) is given by:

[ \text{Average slope} = \frac{f(b) - f(a)}{b - a} ]

Substitute ( a = 1 ) and ( b = 4 ), and compute ( f(1) ) and ( f(4) ):

[ f(1) = \frac{1}{1 + 2} = \frac{1}{3} ] [ f(4) = \frac{4}{4 + 2} = \frac{4}{6} = \frac{2}{3} ]

Now, substitute these values into the formula for average slope:

[ \text{Average slope} = \frac{\frac{2}{3} - \frac{1}{3}}{4 - 1} = \frac{\frac{1}{3}}{3} = \frac{1}{9} ]

Therefore, the average or mean slope of the function ( f(x) = \frac{x}{x + 2} ) on the interval ([1, 4]) is ( \frac{1}{9} ).

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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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