Final answer:
The average value of g(x) on the interval [-1,10] is approximately 64.55. This is determined by solving the weighted average equation based on the average values given for the intervals [-5,-1] and [-5,10].
Step-by-step explanation:
The average value of the function g(x) over an interval can be understood as the mean of all function values within that interval. According to the given information, the average value of g(x) on the interval [-5,-1] is 10, and the average value of g(x) on [-5,10] is 50. To determine the average value of g(x) on [-1,10], we need to use the concept of a weighted average since the function has already been averaged over two overlapping intervals.
The interval [-5, -1] spans 4 units (from -5 to -1), and the interval [-5, 10] spans 15 units (from -5 to 10). From this, the interval [-1, 10] spans 11 units (from -1 to 10). Let A be the average value of g(x) on [-1,10]. The total sum of the function values on [-5,10] is the sum of the values on [-5,-1] and the sum of the values on [-1,10]. Mathematically, this can be expressed as:
4 * 10 + 11 * A = 15 * 50
By solving this equation, we find the average value A on [-1,10] as follows:
40 + 11A = 750
11A = 710
A = 64.5454...
Therefore, the average value of g(x) on the interval [-1,10] is approximately 64.55.