Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4 Figure 3 shows a sketch of part of the curve with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

To sketch the graph of $H(τ) = |10e^{-0.25τ} sin(τ)|$, we note that:

• As $τ$ increases, $e^{-0.25τ}$ decreases, resulting in a decay of the amplitude of the sine function.
• The sine function oscillates between -1 and 1, thus $|sin(τ)|$ will oscillate between 0 and 1.

The graph should show oscillations that decrease in height over time, giving a shape with peaks that gradually become lower, resembling a decaying sinusoidal wave.

To find the maximum height of the ball, we evaluate:

$H(τ) = |10e^{-0.25τ}sin(τ)|$

At the maximum point where $H(τ)$ is maximized, we determine: $dH/dτ = 0$ and evaluate it at the turning points found earlier. Substituting $τ = 1.33 ext{ seconds}$ gives:

$H(1.33) = |10e^{-0.25 * 1.33}sin(1.33)|$

This yields an approximate maximum height of about 3.18 metres.

The model $H(τ)$ provides an ideal estimation of the height of the ball, assuming no energy loss due to factors like air resistance or ground absorption. In reality, the height will decrease with each bounce due to dissipative forces that are not accounted for in this function. Therefore, while it can give a rough estimate of maxima, it cannot reliably predict actual bounce timings or heights after the first bounce.