Question

1. a soccer ball is kicked into the air. the following table of values shows the height of the ball above the ground at various times during its flight.

time (s) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
height (m) 0.5 11.78 20.6 26.98 30.9 32.38 31.4 27.98 22.1 13.78 3.0
a) estimate the instantaneous rate of change in the height of the ball at exactly t = 2.0 s using the preceding and following interval method.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. To estimate the instantaneous rate of change at $t = 2.0$s using the preceding and following - interval method, we consider the interval $[1.5,2.5]$.

Step2: Identify the values from the table

When $t_1=1.5$, $h_1 = 26.98$m and when $t_2 = 2.5$, $h_2=32.38$m.

Step3: Calculate the average rate of change

Using the formula $\frac{h_2 - h_1}{t_2 - t_1}$, we substitute the values: $\frac{32.38 - 26.98}{2.5-1.5}=\frac{5.4}{1}=5.4$ m/s.

Answer:

$5.4$ m/s