Question

find the average rate of change of the function over the given interval.$f(t) = 2t^{2} - 3$, $3, 3.1$compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.left endpointright endpoint

Explanation:

Step1: Define average rate formula

The average rate of change of $f(t)$ over $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$.
Here, $a=3$, $b=3.1$, $f(t)=2t^2-3$.

Step2: Calculate $f(3)$

$f(3)=2(3)^2-3=2(9)-3=18-3=15$

Step3: Calculate $f(3.1)$

$f(3.1)=2(3.1)^2-3=2(9.61)-3=19.22-3=16.22$

Step4: Compute average rate of change

$\frac{f(3.1)-f(3)}{3.1-3}=\frac{16.22-15}{0.1}=\frac{1.22}{0.1}=12.2$

Step5: Find derivative for instantaneous rate

$f'(t)=\frac{d}{dt}(2t^2-3)=4t$

Step6: Instantaneous rate at left endpoint

Left endpoint $t=3$: $f'(3)=4(3)=12$

Step7: Instantaneous rate at right endpoint

Right endpoint $t=3.1$: $f'(3.1)=4(3.1)=12.4$

Answer:

Average rate of change: $12.2$
Instantaneous rate at left endpoint ($t=3$): $12$
Instantaneous rate at right endpoint ($t=3.1$): $12.4$
The average rate of change is between the instantaneous rates at the two endpoints, closer to the left endpoint's rate.