Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P. The x-coordinate of P is k. (e) Show that k satisfies the equation $$4k + \sin 4k - 2 = 0.$$ The iterative formula $$x_{n+1} = \frac{1}{2} \left( 2 - \sin 4x_n \right), \quad x_0 = 0.3,$$ is used to find an approximate value for k. (b) Calculate the values of $x_1, x_2, x_3,$ and $x_4$, giving your answers to 4 decimal places. (c) Show that k = 0.277, correct to 3 significant figures.

A-Level Edexcel Maths Pure Trigonometric Equations: Figure 2 shows part of the curve

Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 4

Question 6

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Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P. The x-coordinate of P ... show full transcript

Worked Solution & Example Answer:Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 4

Step 1

Show that k satisfies the equation

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Answer

To find the value of k satisfying the equation, we first take the derivative of the given function:

$\frac{dy}{dx} = 2 \tan 2x + (2x - 1) 2 \sec^2 2x$

Setting this equal to zero to find critical points:

$2 \tan 2x + (2x - 1) 2 \sec^2 2x = 0$

Rearranging gives:

$2 \tan 2x + 4x\sec^2 2x - 2 = 0$

This leads to the equation:

$4k + \sin 4k - 2 = 0$

which shows k satisfies the required condition.

Step 2

Calculate the values of x1, x2, x3, and x4

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Answer

Using the iterative formula with the initial value:

For (n=0:) $x_1 = \frac{1}{2} \left( 2 - \sin(4 \times 0.3) \right) \approx 0.2670$
For (n=1:) $x_2 = \frac{1}{2} \left( 2 - \sin(4 \times 0.2670) \right) \approx 0.2809$
For (n=2:) $x_3 = \frac{1}{2} \left( 2 - \sin(4 \times 0.2809) \right) \approx 0.2746$
For (n=3:) $x_4 = \frac{1}{2} \left( 2 - \sin(4 \times 0.2746) \right) \approx 0.2774$

Thus, the calculated values are:

(x_1 \approx 0.2670)
(x_2 \approx 0.2809)
(x_3 \approx 0.2746)
(x_4 \approx 0.2774)

Step 3

Show that k = 0.277, correct to 3 significant figures

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Answer

To show that k = 0.277 correct to three significant figures, we analyze the result from our previous iteration:

$x_4 \approx 0.2774$

Rounding this value to three significant figures gives:

$k = 0.277$

This confirms the required accuracy.

Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 4

Worked Solution & Example Answer:Figure 2 shows part of the curve with equation $$y = (2x - 1) \tan 2x, \quad 0 < x < \frac{\pi}{4}$$ The curve has a minimum at the point P - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 4

Show that k satisfies the equation

Calculate the values of x1, x2, x3, and x4

Show that k = 0.277, correct to 3 significant figures

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