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Given y = 3√(x - 6x + 4), x > 0 (a) find ∫ y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

Question 4

Given y = 3√(x - 6x + 4), x > 0 (a) find ∫ y dx, simplifying each term. (b) (i) Find dy/dx (ii) Hence find the value of x such that dy/dx = 0.

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

Step 1

find ∫ y dx, simplifying each term.

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Answer

To evaluate the integral of y, first rewrite y clearly:

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

Now compute the integral:

$\int y \, dx = \int 3\sqrt{4 - 6x} \, dx$

To perform the integration, let's use substitution:

Let ( u = 4 - 6x ) Then, ( du = -6 , dx \rightarrow dx = -\frac{1}{6}du )

Substituting gives:

$\int 3\sqrt{u} \left(-\frac{1}{6}du\right) = -\frac{1}{2} \int u^{1/2} \, du$

Now, integrating:

$= -\frac{1}{2} \left(\frac{u^{3/2}}{3/2}\right) + C = -\frac{1}{3} u^{3/2} + C$

Substituting back for u:

$= -\frac{1}{3} (4 - 6x)^{3/2} + C$

The final result of the integral is:

$\int y \, dx = -\frac{1}{3} (4 - 6x)^{3/2} + C$

Step 2

Find dy/dx

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Answer

To find ( \frac{dy}{dx} ), we can differentiate y with respect to x:

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

Using the chain rule:

$\frac{dy}{dx} = 3 \cdot \frac{1}{2}(4 - 6x)^{-1/2} \cdot (-6)$

Simplifying further:

$\frac{dy}{dx} = -\frac{9}{\sqrt{4 - 6x}}$

Step 3

Hence find the value of x such that dy/dx = 0.

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Answer

Setting the derivative ( \frac{dy}{dx} ) equal to 0:

$-\frac{9}{\sqrt{4 - 6x}} = 0$

The above equation does not yield any real value for x since a square root cannot equal zero in the denominator. Therefore, it cannot be solved by setting ( \frac{dy}{dx} = 0 ). This indicates that there are no critical points where the slope/fall of the curve y is flat. However, if you set the numerator conditionally (i.e., 9 = 0), it holds no real solution. The function has no horizontal tangents; thus, no value of x exists such that ( \frac{dy}{dx} = 0 ).

Given y = 3√(x - 6x + 4), x > 0 (a) find ∫ y dx, simplifying each term - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 1

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

find ∫ y dx, simplifying each term.

Find dy/dx

Hence find the value of x such that dy/dx = 0.

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