Final answer:
To find the average value of the function f(x) = x + x⁴ on the interval [−5,4], integrate the function over this range and divide the result by the interval length, which is 9.
Step-by-step explanation:
To find the average value of the function f(x) = x + x⁴ over the interval [−5,4], you use the formula for the average value of a function over an interval [a, b], which is:
\[\frac{1}{b - a}\int_{a}^{b} f(x) dx\]
First, integrate the function from −5 to 4:
\[\int_{-5}^{4} (x + x⁴) dx\] = \left[ \frac{x^2}{2} + \frac{x^5}{5} \right]_{-5}^{4}
Next, evaluate the integral at the limits of integration:
\[= \left( \frac{4^2}{2} + \frac{4^5}{5} \right) - \left( \frac{(-5)^2}{2} + \frac{(-5)^5}{5} \right)\]
Now, calculate this to find the integral's value, and divide the result by the length of the interval, which is 4 - (-5) = 9.
Finally, you get the average value by applying the formula:
\[\frac{1}{9}(\text{Value of Integral})\]
Remember to perform the calculations step by step to get the final value for the average.