# How do you find the average rate of change of # f(x) = x^2+x+3# from [1,x]?

The average rate of change of a function

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To find the average rate of change of (f(x) = x^2 + x + 3) over the interval ([1, x]), where (x) is a variable, you would use the formula:

[ \text{Average Rate of Change} = \frac{f(x) - f(1)}{x - 1} ]

Plug in the function values for (f(x)) and (f(1)):

[ f(x) = x^2 + x + 3 ] [ f(1) = 1^2 + 1 + 3 = 5 ]

Substitute these values into the formula:

[ \text{Average Rate of Change} = \frac{x^2 + x + 3 - 5}{x - 1} ]

Simplify:

[ \text{Average Rate of Change} = \frac{x^2 + x - 2}{x - 1} ]

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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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