# Suppose #f(x)=x^4-4x^2+2#. What is the average rate of change of the function on the interval [1.9, 2.1]?

The following yields the average rate of change:

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To find the average rate of change of the function ( f(x) = x^4 - 4x^2 + 2 ) on the interval ([1.9, 2.1]), we first need to evaluate the function at the endpoints of the interval.

[ f(1.9) = (1.9)^4 - 4(1.9)^2 + 2 ] [ = 5.8321 - 14.44 + 2 ] [ = -6.6079 ]

[ f(2.1) = (2.1)^4 - 4(2.1)^2 + 2 ] [ = 9.261 - 17.64 + 2 ] [ = -6.379 ]

Then, we can use these values to calculate the average rate of change using the formula:

[ \text{Average rate of change} = \frac{f(2.1) - f(1.9)}{2.1 - 1.9} ]

Substituting the values we found earlier:

[ \text{Average rate of change} = \frac{-6.379 - (-6.6079)}{2.1 - 1.9} ] [ = \frac{-6.379 + 6.6079}{0.2} ] [ = \frac{0.2289}{0.2} ] [ = 1.1445 ]

Therefore, the average rate of change of the function on the interval ([1.9, 2.1]) is approximately (1.1445).

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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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