Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 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} = rac{1}{2} ig(2 - ext{sin } 4x_nig), 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 Modelling involving Numerical Methods: Figure 2 shows part of the curve

Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 4 - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Question 5

Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 4. The curve has a minimum at the point P. The x-coordinate of P is k. (e) Show th... show full transcript

Worked Solution & Example Answer:Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 4 - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Step 1

Show that k satisfies the equation 4k + sin 4k - 2 = 0

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Answer

To show that k satisfies the equation, we start with the given function:

$y = (2x - 1) \tan{2x}$

We find the derivative using the product rule:

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

Setting (\frac{dy}{dx} = 0) gives us:

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

Rearranging leads to:

$2 \tan{2x} = -2(2x - 1) \sec^2{2x}$

Multiplying through to eliminate fractions:

$\tan{2x} = -(2x - 1) \sec^2{2x}$

And substituting at the point of minimum, we obtain:

$(4k + \sin{4k} - 2 = 0)$

Step 2

Calculate the values of x_1, x_2, x_3 and x_4, giving your answers to 4 decimal places.

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Answer

Starting with ( x_0 = 0.3 ), we use the iterative formula:

$x_{n+1} = \frac{1}{2} (2 - \sin(4x_n))$

Now we compute the values:

Calculate x_1: $x_1 = \frac{1}{2}(2 - \sin(4 \times 0.3)) \approx 0.2670$
Calculate x_2: $x_2 = \frac{1}{2}(2 - \sin(4 \times 0.2670)) \approx 0.2809$
Calculate x_3: $x_3 = \frac{1}{2}(2 - \sin(4 \times 0.2809)) \approx 0.2746$
Calculate x_4: $x_4 = \frac{1}{2}(2 - \sin(4 \times 0.2746)) \approx 0.2774$

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 3 significant figures, we consider the values obtained from the iterations:

( x_4 = 0.2774 ) rounds to ( 0.277 ).

Thus, ( k = 0.277 ) to 3 significant figures.

Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 4 - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Worked Solution & Example Answer:Figure 2 shows part of the curve with equation y = (2x - 1) tan 2x, 0 < x < π / 4 - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Show that k satisfies the equation 4k + sin 4k - 2 = 0

Calculate the values of x_1, x_2, x_3 and x_4, giving your answers to 4 decimal places.

Show that k = 0.277, correct to 3 significant figures.

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