A-Level Edexcel Maths Pure Integration: The curve C has equation y = \f

The curve C has equation y = \frac{x}{9+x^2} - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 6

Question 6

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The curve C has equation y = \frac{x}{9+x^2}. Use calculus to find the coordinates of the turning points of C. Given that y = (1 + e^{x})^{\frac{3}{2}}, find the ... show full transcript

Worked Solution & Example Answer:The curve C has equation y = \frac{x}{9+x^2} - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 6

Step 1

Find the coordinates of the turning points of C

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Answer

To find the turning points, we first need to compute the first derivative of the function given by $y = \frac{x}{9 + x^2}.$

Using the quotient rule, we find:

$\frac{dy}{dx} = \frac{(9 + x^2)(1) - (x)(2x)}{(9 + x^2)^2} = \frac{9 + x^2 - 2x^2}{(9 + x^2)^2} = \frac{9 - x^2}{(9 + x^2)^2}.$

Setting the derivative equal to zero to find the critical points:

$9 - x^2 = 0 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3.$

Next, we find the corresponding y-values:

When (x = 3:\n y = \frac{3}{9 + 3^2} = \frac{3}{18} = \frac{1}{6}.$$

When (x = -3:\n y = \frac{-3}{9 + (-3)^2} = \frac{-3}{18} = -\frac{1}{6}.\n Thus, the turning points of the curve are at ((3, \frac{1}{6})) and ((-3, -\frac{1}{6})).

Step 2

Find the value of \frac{dy}{dx} at x = \frac{1}{2} \ln 3

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Answer

Given the function $y = (1 + e^{x})^{\frac{3}{2}},$ we will compute the derivative:

Using the chain rule:

$\frac{dy}{dx} = \frac{3}{2}(1 + e^{x})^{\frac{1}{2}} \cdot e^{x}.$

Now, we substitute (x = \frac{1}{2} \ln 3:\n e^{x} = e^{\frac{1}{2} \ln 3} = \sqrt{3}.$$

Substituting this in:

$\frac{dy}{dx} = \frac{3}{2}(1 + \sqrt{3})^{\frac{1}{2}} \cdot \sqrt{3}.$

Now simplifying,

$\frac{dy}{dx} = \frac{3 \sqrt{3}}{2(1 + \sqrt{3})^{\frac{1}{2}}}.$

This gives us the final answer.

The curve C has equation y = \frac{x}{9+x^2} - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 6

Worked Solution & Example Answer:The curve C has equation y = \frac{x}{9+x^2} - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 6

Find the coordinates of the turning points of C

Find the value of \frac{dy}{dx} at x = \frac{1}{2} \ln 3

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