The curve with equation $y = f(x)$ has a turning point $P$. (a) Find the exact coordinates of $P$. The equation $f(x) = 0$ has a root between $x = 0.25$ and $x = 0.3$. (b) Use the iterative formula $$x_{n+1} = \frac{1}{3} e^{-x_n}$$ with $x_0 = 0.25$ to find, to 4 decimal places, the values of $x_1$, $x_2$, and $x_3$. (c) By choosing a suitable interval, show that a root of $f(x) = 0$ is $x = 0.2576$ correct to 4 decimal places.

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The curve with equation $y = f(x)$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 1 - 2009 - Paper 2

Question 1

The curve with equation $y = f(x)$ has a turning point $P$. (a) Find the exact coordinates of $P$. The equation $f(x) = 0$ has a root between $x = 0.25$ and $x... show full transcript

Worked Solution & Example Answer:The curve with equation $y = f(x)$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 1 - 2009 - Paper 2

Step 1

Find the exact coordinates of P.

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Answer

To find the turning point $P$ , we need to find the first derivative of the function and set it to zero:

The first derivative is given by: $f'(x) = 3e^x + 3xe^x$
Setting $f'(x) = 0$ leads to: $3e^x(1 + x) = 0$
Thus, we can solve for $x$ :

This gives us: [x = -1]
Now, substituting $x = -1$ back into the original equation to find the coordinates: $f(-1) = 3(-1)e^{-1} - 1 = -\frac{3}{e} - 1$
Thus, the coordinates of point $P$ are ((-1, -\frac{3}{e} - 1)).

Step 2

Use the iterative formula

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Answer

To find the values of $x_1$ , $x_2$ , and $x_3$ using the iterative formula:

Given: $x_{n+1} = \frac{1}{3} e^{-x_n}$

Starting with:

$x_0 = 0.25$

Calculate $x_1$ : $x_1 = \frac{1}{3} e^{-0.25} \approx 0.2596$
Calculate $x_2$ : $x_2 = \frac{1}{3} e^{-0.2596} \approx 0.2571$
Calculate $x_3$ : $x_3 = \frac{1}{3} e^{-0.2571} \approx 0.2578$

Step 3

By choosing a suitable interval, show that a root of f(x) = 0 is x = 0.2576 correct to 4 decimal places.

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Answer

To demonstrate that there is a root in the interval $(0.25755, 0.25765)$ :

Evaluate $f(0.25755)$ and $f(0.25765)$ :

For $f(0.25755)$ : $f(0.25755) = 0.000379...$
For $f(0.25765)$ : $f(0.25765) = 0.000109...$

Since $f(0.25755)$ and $f(0.25765)$ have opposite signs, and the function is continuous, by the Intermediate Value Theorem, there is at least one root in the interval $(0.25755, 0.25765)$ . Thus, we can conclude:

$x = 0.2576$

is correct to 4 decimal places.

The curve with equation $y = f(x)$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 1 - 2009 - Paper 2

Worked Solution & Example Answer:The curve with equation $y = f(x)$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 1 - 2009 - Paper 2

Find the exact coordinates of P.

Use the iterative formula

By choosing a suitable interval, show that a root of f(x) = 0 is x = 0.2576 correct to 4 decimal places.

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