A-Level AQA Maths: Pure 7.5 Implicit Differentiation: Find the coordinates of the stat

Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Question 6

Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$

Worked Solution & Example Answer:Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Step 1

Select appropriate technique to differentiate

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Answer

To find the stationary points, we first need to differentiate the given equation implicitly, treating $y$ as a function of $x$ . We start by differentiating both sides of the equation:

$2(x+y-2)\left(\frac{d}{dx}(x+y-2)\right) = e^y \frac{dy}{dx}$

Applying the chain rule to the left side gives us:

$2(x+y-2)\left(1 + \frac{dy}{dx}\right) = e^y \frac{dy}{dx}$

Step 2

Differentiate term involving $e^y$ correctly

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Answer

Rearranging the differentiation leads to:

$2(x+y-2) + 2(x+y-2)\frac{dy}{dx} = e^y \frac{dy}{dx}$

Step 3

Set $rac{dy}{dx} = 0$

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Answer

For stationary points, we set the derivative $rac{dy}{dx} = 0$ :

$2(x+y-2) = 0$

Step 4

Solve for $y$

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Answer

This simplifies to:

$x + y - 2 = 0 \Rightarrow y = 2 - x$

Step 5

Eliminate $x$ or $y$ from the equation of the curve

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Answer

Now, substituting $y = 2 - x$ back into the original curve's equation:

$(x + (2 - x) - 2)^2 = e^{(2 - x)} - 1$

This simplifies to:

$0 = e^{(2 - x)} - 1 \Rightarrow e^{(2 - x)} = 1$

Step 6

Obtain correct $y$

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Answer

Taking the natural logarithm, we find:

$2 - x = 0 \Rightarrow x = 2$

Substituting $x = 2$ back into $y = 2 - x$ , gives:

$y = 2 - 2 = 0$

Step 7

Obtain correct $x$

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Answer

Thus, the coordinates of the stationary point are:

Coordinates: $(2, 0)$

Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Worked Solution & Example Answer:Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Select appropriate technique to differentiate

Differentiate term involving $e^y$ correctly

Set $rac{dy}{dx} = 0$

Solve for $y$

Eliminate $x$ or $y$ from the equation of the curve

Obtain correct $y$

Obtain correct $x$

Join the A-Level students using SimpleStudy...

Other A-Level Maths: Pure topics to explore

Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Worked Solution & Example Answer:Find the coordinates of the stationary point of the curve with equation $(x+y-2)^2 = e^y - 1$ - AQA - A-Level Maths: Pure - Question 6 - 2018 - Paper 2

Select appropriate technique to differentiate

Differentiate term involving $e^y$ correctly

Set $ rac{dy}{dx} = 0$

Solve for $y$

Eliminate $x$ or $y$ from the equation of the curve

Obtain correct $y$

Obtain correct $x$

Join the A-Level students using SimpleStudy...

Other A-Level Maths: Pure topics to explore

Set $rac{dy}{dx} = 0$