Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians. The point $P$, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2, a) find the $x$ coordinate of $P$, giving your answer to 3 significant figures. The curve crosses the $x$-axis at $x = a$, as shown in Figure 2. Given that, to 3 decimal places, $f(4) = 4.274$ and $f(5) = -1.212$ (b) explain why $a$ must lie in the interval $[4, 5]$. Taking $x_0 = 5$ as a first approximation to $a$, apply the Newton-Raphson method once to $f(x)$ to obtain a second approximation to $a$. (c) Show your method and give your answer to 3 significant figures.

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Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2

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Figure-2-shows-a-sketch-of-part-of-the-curve-with-equation-$y-=-f(x)$-where--$f(x)-=-8--ext{sin}-igg(--rac{1}{2}x-igg)---3x-+-9$---$x->-0$--and-$x$-is-measured-in-radians-Edexcel-A-Level Maths Pure-Question 8-2022-Paper 2.png

Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in ... show full transcript

Worked Solution & Example Answer:Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2

Step 1

a) find the $x$ coordinate of $P$, giving your answer to 3 significant figures.

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Answer

To find the $x$ coordinate of point $P$ , we first need to determine the derivative of the function, which gives us the slope of the curve at any point:

$f'(x) = 4 ext{cos} igg( rac{1}{2} x igg) - 3$

To find the local maximum point, we set the derivative equal to zero:

$4 ext{cos} igg( rac{1}{2} x igg) - 3 = 0$

This leads to:

$ext{cos} igg( rac{1}{2} x igg) = rac{3}{4}$

Now we take the inverse cosine:

$rac{1}{2} x = ext{cos}^{-1} igg( rac{3}{4} igg)$

Hence:

$x = 2 ext{cos}^{-1} igg( rac{3}{4} igg)$

Calculating this value yields:

$x ext{ (approx)} = 2 imes 0.722 = 1.444$

Therefore, when rounded to three significant figures:

$x_P = 1.44$

Step 2

b) explain why $a$ must lie in the interval $[4, 5]$.

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Answer

To explain why $a$ must lie in the interval $[4, 5]$ , we evaluate the function values provided:

$f(4) = 4.274 > 0$
$f(5) = -1.212 < 0$

Since $f(4)$ is positive and $f(5)$ is negative, by the Intermediate Value Theorem, there must be at least one root in the interval $[4, 5]$ because the function changes sign. This confirms that $a$ , the point where the curve crosses the x-axis, must lie within this interval.

Step 3

c) apply the Newton-Raphson method once to $f(x)$ to obtain a second approximation to $a$.

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Answer

For the Newton-Raphson method, we will use the formula:

$x_{n+1} = x_n - rac{f(x_n)}{f'(x_n)}$

Given our first approximation $x_0 = 5$ , we calculate:

Evaluate $f(5)$ : $f(5) = -1.212$
Evaluate $f'(5)$ : $f'(5) = 4 ext{cos} igg( rac{1}{2} imes 5 igg) - 3 = 4 ext{cos}(2.5) - 3$ Using a calculator, we find: $f'(5) ext{ (approx)} = 4 imes (-0.801) - 3 ag{approximately} = -6.204$

Now, substituting into the Newton-Raphson formula:

$x_1 = 5 - rac{-1.212}{-6.204}$ $x_1 ext{ (approx)} = 5 + 0.195 = 4.805$

Rounding this to three significant figures, we have:

$x_1 ext{ (approx)} = 4.80$

Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2

Worked Solution & Example Answer:Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2

a) find the $x$ coordinate of $P$, giving your answer to 3 significant figures.

b) explain why $a$ must lie in the interval $[4, 5]$.

c) apply the Newton-Raphson method once to $f(x)$ to obtain a second approximation to $a$.

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Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2

Worked Solution & Example Answer:Figure 2 shows a sketch of part of the curve with equation $y = f(x)$ where $f(x) = 8 ext{sin} igg( rac{1}{2}x igg) - 3x + 9$ $x > 0$ and $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 2