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Given y = 3\sqrt{\:x - 6x + 4}, \: x > 0 (a) find \int y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

Question 4

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Given y = 3\sqrt{\:x - 6x + 4}, \: x > 0 (a) find \int y dx, simplifying each term. (b) (i) Find \frac{dy}{dx} (ii) Hence find the value of x such that \frac{dy}... show full transcript

Worked Solution & Example Answer:Given y = 3\sqrt{\:x - 6x + 4}, \: x > 0 (a) find \int y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

Step 1

find \int y dx, simplifying each term.

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Answer

To find the integral ( \int y dx ), we start with the given equation for y:

$y = 3\sqrt{x - 6x + 4} = 3\sqrt{-5x + 4}$

Next, we will integrate y:

$\int y \, dx = 3 \int \sqrt{-5x + 4} \, dx$

Using the substitution ( u = -5x + 4 ), we find:

$du = -5 \, dx \, \Rightarrow \, dx = -\frac{1}{5} du$

Thus, the integral becomes:

$\int y \, dx = 3 \int \sqrt{u} \left(-\frac{1}{5} du\right)\ = -\frac{3}{5} \int u^{1/2} \, du$

Calculating the integral:

$= -\frac{3}{5} \left( \frac{u^{3/2}}{\frac{3}{2}} \right) + C = -\frac{3}{5} \left( \frac{2}{3} u^{3/2} \right) + C = -\frac{2}{5} u^{3/2} + C$

Substituting back for ( u ):

$= -\frac{2}{5} (-5x + 4)^{3/2} + C$

Step 2

Find \frac{dy}{dx}

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Answer

To determine ( \frac{dy}{dx} ), we will differentiate y:

$y = 3\sqrt{-5x + 4}$

Using the chain rule:

$\frac{dy}{dx} = 3 \cdot \frac{1}{2} (-5x + 4)^{-1/2} \cdot (-5) = -\frac{15}{2\sqrt{-5x + 4}}$

Step 3

Hence find the value of x such that \frac{dy}{dx} = 0

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Answer

For ( \frac{dy}{dx} = 0 ):

Set the expression to zero:

$-\frac{15}{2\sqrt{-5x + 4}} = 0$

Since the numerator must be zero (denominator cannot be zero), we have:

This does not yield any solution for x directly, as -15 does not equal zero. Therefore, there are no values of x that make ( \frac{dy}{dx} = 0 ) in this case.

However, if additional scenarios (like endpoints or limits) are considered, then the function behavior should be evaluated further using the original function's constraints.

Given y = 3\sqrt{\:x - 6x + 4}, \: x > 0 (a) find \int y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

Worked Solution & Example Answer:Given y = 3\sqrt{\:x - 6x + 4}, \: x > 0 (a) find \int y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

find \int y dx, simplifying each term.

Find \frac{dy}{dx}

Hence find the value of x such that \frac{dy}{dx} = 0

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