Question4. The path of a baseball relative to the ground can be modelled by the function $d(t)=-t^{2}+8 t+1$ where $d(t)$ represents the height of the ball in metres after $t$ seconds. a. Find the average rate of change of the ball between 1 and 3 seconds. [4 marks] b. Using the secant method, find the instantaneous rate of change at 2 seconds. [5 marks]

Studdy Solution
Calculate the instantaneous rate of change at $t=2$ using the secant method formula $\frac{d(2+h) - d(2)}{h}$ with $h = 0.01$.
$$\text{Instantaneous rate of change} = \frac{d(2.01) - d(2)}{0.01} = \frac{13.0399 - 13}{0.01} = \frac{0.0399}{0.01} = 3.99 \text{ meters per second}$$
Solution: a. The average rate of change of the ball between 1 and 3 seconds is $4$ meters per second. b. Using the secant method, the instantaneous rate of change at 2 seconds is approximately $3.99$ meters per second.