# How do you find the average rate of change of y with respect to x over the interval [1, 3] given #y=3x^4#?

Use a difference quotient.

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To find the average rate of change of ( y ) with respect to ( x ) over the interval ([1, 3]) for the function ( y = 3x^4 ), you first evaluate the function at the endpoints of the interval to find the corresponding values of ( y ), then use the formula:

[ \text{Average Rate of Change} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1} ]

where ( (x_1, y_1) ) and ( (x_2, y_2) ) are the coordinates of the endpoints of the interval. In this case:

[ x_1 = 1, \quad x_2 = 3 ] [ y_1 = 3(1)^4 = 3, \quad y_2 = 3(3)^4 = 81 ]

So, the average rate of change of ( y ) with respect to ( x ) over the interval ([1, 3]) is:

[ \text{Average Rate of Change} = \frac{81 - 3}{3 - 1} = \frac{78}{2} = 39 ]

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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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