The curve with equation $y = f(x) = 3xe^x - 1$ 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 $x_{n+1} = rac{1}{3} e^{x_n}$ 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 Applications of Differentiation: The curve with equation $y = f(x

The curve with equation $y = f(x) = 3xe^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2

Question 7

The curve with equation $y = f(x) = 3xe^x - 1$ 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 $y = f(x) = 3xe^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2

Step 1

Find the exact coordinates of P.

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Answer

To find the coordinates of the turning point $P$ , we first determine where the derivative of $f(x)$ is zero.

Calculating the first derivative:

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

Setting this to zero:

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

This yields:

eq 0$$ Thus, we can solve: $$1 + x = 0 \implies x = -1$$ We substitute $x = -1$ back into the original function to find $y$: $$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 with $x_0 = 0.25$ to find, to 4 decimal places, the values of $x_1$, $x_2$, and $x_3$.

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Answer

Using the iterative formula:

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

Step 1: Using $x_0 = 0.25$ :

$x_1 = \frac{1}{3} e^{0.25} \approx 0.2596$

Step 2: Using $x_1 = 0.2596$ :

$x_2 = \frac{1}{3} e^{0.2596} \approx 0.2571$

Step 3: Using $x_2 = 0.2571$ :

$x_3 = \frac{1}{3} e^{0.2571} \approx 0.2578$

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 a root lies within $[0.2575, 0.25765]$ , we evaluate:

$f(0.2575) \approx 0.000109$ $f(0.25765) \approx -0.000379$

Since $f(0.2575) > 0$ and $f(0.25765) < 0$ , by the Intermediate Value Theorem, there exists at least one root in the interval $(0.2575, 0.25765)$ . Thus, we conclude that $x = 0.2576$ is correct to 4 decimal places.

The curve with equation $y = f(x) = 3xe^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2

Worked Solution & Example Answer:The curve with equation $y = f(x) = 3xe^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2

Find the exact coordinates of P.

Use the iterative formula with $x_0 = 0.25$ to find, to 4 decimal places, the values of $x_1$, $x_2$, and $x_3$.

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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