Step 1:
To find the average value \( f_{\text{ave}} \) of the function \( f(x) = x^7 \) over the interval \([-1, 1]\), we use the formula for the average value of a function over an interval \([a, b]\):
\[ f_{\text{ave}} = \frac{1}{b-a} \int_a^b f(x) \, dx \]
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