Answer:
Average rate of change is 1.
Step-by-step explanation:
The average rate of change of function
over the intervals
is given by the formula,
.
Rewriting the formula according to given data,
.
Given the interval as
. Hence value of a and b is a = -4 and b = 5.
Now calculate
and
as follows.
Calculation for
,
![g\left(t\right)=-\left(t-1\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%28t%5Cright%29%3D-%5Cleft%28t-1%5Cright%29%5E2%2B5)
Replace t as a,
![g\left(a\right)=-\left(a-1\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%28a%5Cright%29%3D-%5Cleft%28a-1%5Cright%29%5E2%2B5)
Substituting the value,
![g\left(-4\right)=-\left(-4-1\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%28-4%5Cright%29%3D-%5Cleft%28-4-1%5Cright%29%5E2%2B5)
![g\left(-4\right)=-\left(-5\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%28-4%5Cright%29%3D-%5Cleft%28-5%5Cright%29%5E2%2B5)
![g\left(-4\right)=-25+5](https://tex.z-dn.net/?f=g%5Cleft%28-4%5Cright%29%3D-25%2B5)
![g\left(-4\right)=-20](https://tex.z-dn.net/?f=g%5Cleft%28-4%5Cright%29%3D-20)
Calculation for
,
![g\left(b\right)=-\left(b-1\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%28b%5Cright%29%3D-%5Cleft%28b-1%5Cright%29%5E2%2B5)
Substituting the value,
![g\left(5\right)=-\left(5-1\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%285%5Cright%29%3D-%5Cleft%285-1%5Cright%29%5E2%2B5)
![g\left(5\right)=-\left(4\right)^2+5](https://tex.z-dn.net/?f=g%5Cleft%285%5Cright%29%3D-%5Cleft%284%5Cright%29%5E2%2B5)
![g\left(5\right)=-16+5](https://tex.z-dn.net/?f=g%5Cleft%285%5Cright%29%3D-16%2B5)
![g\left(5\right)=-11](https://tex.z-dn.net/?f=g%5Cleft%285%5Cright%29%3D-11)
Now substituting the value,
.
Simplifying,
.
.
.
Hence average rate of change of the given function
over the given interval
is 1.