g(t)=-(t - 1)^{2}+5\nwhat is the average rate of change of g over the interval -4 ≤ t ≤ 5?
g(t)=-(t - 1)^{2}+5\nwhat is the average rate of change of g over the interval -4 ≤ t ≤ 5?
Answer
Explanation:
Step1: Recall the formula for average rate of change
The average rate of change of a function (y = g(t)) over the interval ([a,b]) is given by (\frac{g(b)-g(a)}{b - a}). Here, (a=-4) and (b = 5).
Step2: Calculate (g(-4))
Substitute (t=-4) into (g(t)=-(t - 1)^{2}+5). [ \begin{align*} g(-4)&=-(-4 - 1)^{2}+5\ &=-( - 5)^{2}+5\ &=-25 + 5\ &=-20 \end{align*} ]
Step3: Calculate (g(5))
Substitute (t = 5) into (g(t)=-(t - 1)^{2}+5). [ \begin{align*} g(5)&=-(5 - 1)^{2}+5\ &=-16+5\ &=-11 \end{align*} ]
Step4: Calculate the average rate of change
Using the formula (\frac{g(b)-g(a)}{b - a}), with (a=-4), (b = 5), (g(-4)=-20) and (g(5)=-11) [ \begin{align*} \frac{g(5)-g(-4)}{5-(-4)}&=\frac{-11-(-20)}{5 + 4}\ &=\frac{-11 + 20}{9}\ &=\frac{9}{9}\ &=1 \end{align*} ]
Answer:
(1)