# What is the average value of the function #F(x) = e ^ -x # on the interval #[0,4]#?

plot{e^{-x}x^2 [-10, 10, -5, 5]}

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To find the average value of a function ( F(x) ) on the interval ([a, b]), you use the formula:

[ \text{Average Value} = \frac{1}{b - a} \int_{a}^{b} F(x) , dx ]

For ( F(x) = e^{-x} ) on the interval ([0, 4]):

[ \text{Average Value} = \frac{1}{4 - 0} \int_{0}^{4} e^{-x} , dx ]

[ = \frac{1}{4} \left[-e^{-x}\right]_{0}^{4} ]

[ = \frac{1}{4} \left(-e^{-4} + e^0\right) ]

[ = \frac{1}{4} \left(-\frac{1}{e^4} + 1\right) ]

[ \approx \frac{1 - e^{-4}}{4} ]

[ \approx \frac{1 - \frac{1}{e^4}}{4} ]

[ \approx \frac{e^4 - 1}{4e^4} ]

[ \approx 0.3293 ]

So, the average value of ( F(x) = e^{-x} ) on the interval ([0, 4]) is approximately ( 0.3293 ).

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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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