The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$. (a) Find the exact coordinates of $P$. (b) The equation $f(x) = 0$ has a root between $x = 0.25$ and $x = 0.3$. (c) 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.

A-Level Edexcel Maths Pure Modelling with Trigonometric Functions: The curve with equation $y = f(x

The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2

Question 8

The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$. (a) Find the exact coordinates of $P$. (b) The equation $f(x) = 0$ has a root between $x =... show full transcript

Worked Solution & Example Answer:The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2

Step 1

Find the exact coordinates of P.

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Answer

To find the turning point, we first calculate the first derivative of the function:

$f'(x) = 3e^x + 3xe^x$

Setting this equal to zero to locate the turning point:

$3e^x(1 + x) = 0$

Since $e^x$ is never zero, we can solve for:

$1 + x = 0 \\ x = -1$

Now substituting $x = -1$ into the original function to find $y$ :

$f(-1) = 3(-1)e^{-1} - 1 = -\frac{3}{e} - 1$

Thus, the coordinates of $P$ are $(-1, -\frac{3}{e} - 1)$ .

Step 2

Use the iterative formula with $x_0 = 0.25$

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Answer

We start with $x_0 = 0.25$ .

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

Thus, the values are:

$x_1 \approx 0.2596$
$x_2 \approx 0.2571$
$x_3 \approx 0.2578$ .

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

Choosing the interval $(0.2575, 0.25765)$ :

Calculate $f(0.2575) = 3(0.2575)e^{0.2575} - 1 \approx 0.000379$ (positive value).
Calculate $f(0.25765) = 3(0.25765)e^{0.25765} - 1 \approx -0.000109$ (negative value).

Since there is a change of sign in the interval $(0.2575, 0.25765)$ , by the Intermediate Value Theorem, there exists a root in this interval. Thus, $x = 0.2576$ is accurate to 4 decimal places.

The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2

Worked Solution & Example Answer:The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2

Find the exact coordinates of P.

Use the iterative formula with $x_0 = 0.25$

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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