Sure, let's solve the problem step by step.
We have two intervals with given average rates of change:
1. For the interval [tex]\(-3 \leq x \leq 0\)[/tex]:
- The average rate of change is 1.
- The interval length is [tex]\(0 - (-3) = 3\)[/tex].
2. For the interval [tex]\(0 \leq x \leq 3\)[/tex]:
- The average rate of change is 5.
- The interval length is [tex]\(3 - 0 = 3\)[/tex].
To find the total change in [tex]\(y\)[/tex] for each of the intervals:
1. From [tex]\(-3\)[/tex] to [tex]\(0\)[/tex]:
- Change in [tex]\(y\)[/tex] is given by the formula: (rate of change) [tex]\(\times\)[/tex] (length of interval).
- Therefore, the change in [tex]\(y\)[/tex] is [tex]\(1 \times 3 = 3\)[/tex].
2. From [tex]\(0\)[/tex] to [tex]\(3\)[/tex]:
- Change in [tex]\(y\)[/tex] is: [tex]\(5 \times 3 = 15\)[/tex].
Now, combine these changes to find the total change in [tex]\(y\)[/tex] from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex]:
- Total change in [tex]\(y\)[/tex] = [tex]\(3 + 15 = 18\)[/tex].
Next, determine the combined interval length from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex]:
- Length of the combined interval [tex]\( = 3 - (-3) = 6\)[/tex].
Finally, to find the average rate of change over the combined interval from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], we use the total change in [tex]\(y\)[/tex] and the total interval length:
- Average rate of change = (Total change in [tex]\(y\)[/tex]) [tex]\(/\)[/tex] (Length of combined interval).
- Therefore, the average rate of change = [tex]\(18 / 6 = 3.0\)[/tex].
So, the average rate of change of [tex]\(g\)[/tex] on the interval [tex]\( -3 \leq x \leq 3 \)[/tex] is [tex]\( 3.0 \)[/tex].
