12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$. (b) Sketch the graph of $H(t)$ where $H(t) = |10e^{-0.25t} \sin x| \, \text{for} \, t > 0$ showing the long-term behaviour of this curve. (c) The function $H(t)$ is used to model the height, in metres, of a ball above the ground $t$ seconds after it has been kicked. Using this model, find (c) the maximum height of the ball above the ground between the first and second bounce. (d) Explain why this model should not be used to predict the time of each bounce.

A-Level Edexcel Maths Pure Applications of Differentiation: 12. (a) Show that the x coordin

12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$ - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Question 13

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12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$. (b) Sketch the graph of $H(t)$ w... show full transcript

Worked Solution & Example Answer:12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$ - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Step 1

Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$.

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Answer

To find the turning points, we need to compute the derivative of the function. Given that

$f(x) = 10e^{-0.25x} \sin x$

we apply the product rule:

$f'(x) = 10e^{-0.25x} \cos x + 2.5e^{-0.25x} \sin x.$

Setting $f'(x) = 0$ , we have

\Rightarrow \cos x + 0.25 \sin x = 0 \ \Rightarrow \tan x = -4. $$ Since the x-coordinates for turning points are where $ an x = 4$, we need to focus on the principal solution of this trigonometric equation.

Step 2

Sketch the graph of $H(t)$ against $t$ where $H(t) = |10e^{-0.25t} \sin x|$ for $t > 0$ showing the long-term behaviour of this curve.

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Answer

The graph of $H(t)$ will exhibit oscillating behaviour due to the sine function, while the exponential decay term ensures that the amplitude of oscillation decreases over time. As $t$ increases, $|10e^{-0.25t}|$ causes the peaks of the waves (from the sine function) to get closer to the x-axis, indicating that the height of the ball will eventually approach zero. We should illustrate at least two complete cycles to show the behaviour clearly.

Step 3

Find the maximum height of the ball above the ground between the first and second bounces.

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Answer

To find the maximum height between the first and second bounces, we first need to determine when the sine function achieves its maximum value, which is 1. The equation to solve is:

When substituting $t = 0$, we have: \begin{align*} H(0) & = |10e^{-0.25(0)} \cdot 1| \ & = |10| \ & = 10 \text{ metres.} \end{align*} Therefore, the maximum height is 10 metres.

Step 4

Explain why this model should not be used to predict the time of each bounce.

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Answer

This model assumes a consistent pattern of bounces with defined heights that rely solely on the functions provided. However, real-world factors such as air resistance, variations in the ball's material, and external conditions can drastically affect the actual height and time of each bounce. Hence, while the model provides a theoretical framework, it can't accurately predict real-world behavior over successive bounces.

12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$ - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Worked Solution & Example Answer:12. (a) Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$ - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 2

Show that the x coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation $\tan x = 4$.

Sketch the graph of $H(t)$ against $t$ where $H(t) = |10e^{-0.25t} \sin x|$ for $t > 0$ showing the long-term behaviour of this curve.

Find the maximum height of the ball above the ground between the first and second bounces.

Explain why this model should not be used to predict the time of each bounce.

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