Figure 2 shows part of the curve with equation $y = (2x - 1) \tan(2x)$, $0 < x < \frac{\pi}{4}$. 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 $x_{n+1} = \frac{1}{2} \left( 2 - \sin(4x_n) \right), \, x_0 = 0.3,$ 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 Solving Equations: Figure 2 shows part of the curve

Figure 2 shows part of the curve with equation $y = (2x - 1) \tan(2x)$, $0 < x < \frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Question 5

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Figure 2 shows part of the curve with equation $y = (2x - 1) \tan(2x)$, $0 < x < \frac{\pi}{4}$. 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 $y = (2x - 1) \tan(2x)$, $0 < x < \frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Step 1

Show that k satisfies the equation

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Answer

To find the value of k that satisfies the equation, we need to start with the derivative of the function:

Differentiate the equation: Using the product rule:

$\frac{dy}{dx} = (2 \tan(2x) + (2x - 1) \cdot 2 \sec^2(2x))$
Set the derivative to zero: We want to find the critical points:

$2 \tan(2x) + (2x - 1) \cdot 2 \sec^2(2x) = 0$
Substitute to eliminate fractions: After some algebra, we can rewrite this to find the relation:

$4k + \sin(4k) - 2 = 0$ This shows that the x-coordinate k satisfies the equation needed.

Step 2

Calculate the values of $x_1$, $x_2$, $x_3$, and $x_4$

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Answer

We use the iterative formula starting with $x_0 = 0.3$ :

Calculate $x_1$ : $x_1 = \frac{1}{2} \left( 2 - \sin(4 \cdot 0.3) \right) = 0.2769$
Calculate $x_2$ : $x_2 = \frac{1}{2} \left( 2 - \sin(4 \cdot 0.2769) \right) = 0.2809$
Calculate $x_3$ : $x_3 = \frac{1}{2} \left( 2 - \sin(4 \cdot 0.2809) \right) = 0.2746$
Calculate $x_4$ : $x_4 = \frac{1}{2} \left( 2 - \sin(4 \cdot 0.2746) \right) = 0.2774$

Thus, the results are:

$x_1 = 0.2769$
$x_2 = 0.2809$
$x_3 = 0.2746$
$x_4 = 0.2774$

Step 3

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

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Answer

To show the value of k:

From our calculations above, we found that $x_4 = 0.2774$ .
Rounding this value to three significant figures gives us:

$k = 0.277$

Thus, $k$ is shown to be 0.277 when rounded to three significant figures.

Figure 2 shows part of the curve with equation $y = (2x - 1) \tan(2x)$, $0 < x < \frac{\pi}{4}$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

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

Show that k satisfies the equation

Calculate the values of $x_1$, $x_2$, $x_3$, and $x_4$

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

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