October 2020 Paper 2 Q7

EdexcelCurrent spec10 marksDifferentiation Numerical Methods

Figure 1: curve C for x > 0, decreasing steeply from near the positive y-axis to a minimum point P above the x-axis and then increasing — Figure 1

Figure 1 shows a sketch of the curve \(C\) with equation\[y = \frac{4x^2 + x}{2\sqrt{x}} - 4\ln x \qquad x \gt 0\]

(a) Show that\[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}}\] (4)

The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).

(b) Show that the \(x\) coordinate of \(P\) is a solution of\[x = \left(\frac{4}{3} - \frac{\sqrt{x}}{12}\right)^{\frac{2}{3}}\] (3)

(c) Use the iteration formula\[x_{n+1} = \left(\frac{4}{3} - \frac{\sqrt{x_n}}{12}\right)^{\frac{2}{3}} \qquad \text{with } x_1 = 2\]to find

(i) the value of \(x_2\) to 5 decimal places,

(ii) the \(x\) coordinate of \(P\) to 5 decimal places. (3)

Mark scheme (a)

Scheme	Marks	AO
\(\ln x \to \dfrac{1}{x}\)	B1	1.1a
Method to differentiate \(\dfrac{4x^2 + x}{2\sqrt{x}}\) − see notes	M1	1.1b
E.g. \(2 \times \dfrac{3}{2}x^{\frac{1}{2}} + \dfrac{1}{2} \times \dfrac{1}{2}x^{-\frac{1}{2}}\)	A1	1.1b
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\sqrt{x} + \dfrac{1}{4\sqrt{x}} - \dfrac{4}{x} = \dfrac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}}\) *	A1*	2.1
	(4)

Notes

B1: Differentiates \(\ln x \to \dfrac{1}{x}\) seen or implied

M1: Correct method to differentiate \(\dfrac{4x^2 + x}{2\sqrt{x}}\):
Look for \(\dfrac{4x^2 + x}{2\sqrt{x}} \to \ldots x^{\frac{3}{2}} + \ldots x^{\frac{1}{2}}\) being then differentiated to \(Px^{\frac{1}{2}} + \ldots\) or \(\ldots + Qx^{-\frac{1}{2}}\)
Alternatively uses the quotient rule on \(\dfrac{4x^2 + x}{2\sqrt{x}}\).
Condone slips but if rule is not quoted expect \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right) = \dfrac{2\sqrt{x}(Ax + B) - (4x^2 + x)Cx^{-\frac{1}{2}}}{(2\sqrt{x})^2}\ (A, B, C \gt 0)\)
But a correct rule may be implied by their \(u, v, u^{\prime}, v^{\prime}\) followed by applying \(\dfrac{vu^{\prime} - uv^{\prime}}{v^2}\) etc.
Alternatively uses the product rule on \((4x^2 + x)(2\sqrt{x})^{-1}\)
Condone slips but expect \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right) = Ax^{-\frac{1}{2}}(Bx + C) + D(4x^2 + x)x^{-\frac{3}{2}}\ (A, B, C \gt 0)\)
In general condone missing brackets for the M mark. If they quote \(u = 4x^2 + x\) and \(v = 2\sqrt{x}\) and don’t make the differentiation easier, they can be awarded this mark for applying the correct rule. Also allow this mark if they quote the correct quotient rule but only have \(v\) rather than \(v^2\) in the denominator.

A1: Correct differentiation of \(\dfrac{4x^2 + x}{2\sqrt{x}}\) although may not be simplified.
Examples: \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right) = \dfrac{2\sqrt{x}(8x + 1) - (4x^2 + x)x^{-\frac{1}{2}}}{(2\sqrt{x})^2},\ \dfrac{1}{2}x^{-\frac{1}{2}}(8x + 1) - \dfrac{1}{4}(4x^2 + x)x^{-\frac{3}{2}},\ 2 \times \dfrac{3}{2}x^{\frac{1}{2}} + \dfrac{1}{2} \times \dfrac{1}{2}x^{-\frac{1}{2}}\)

A1*: Obtains \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}}\) via \(3\sqrt{x} + \dfrac{1}{4\sqrt{x}} - \dfrac{4}{x}\) or a correct application of the quotient or product rule and with sufficient working shown to reach the printed answer.
There must be no errors e.g. missing brackets.

Mark scheme (b)

Scheme	Marks	AO
\(12x^2 + x - 16\sqrt{x} = 0 \Rightarrow 12x^{\frac{3}{2}} + x^{\frac{1}{2}} - 16 = 0\)	M1	1.1b
E.g. \(12x^{\frac{3}{2}} = 16 - \sqrt{x}\)	dM1	1.1b
\(x^{\frac{3}{2}} = \dfrac{4}{3} - \dfrac{\sqrt{x}}{12} \Rightarrow x = \left(\dfrac{4}{3} - \dfrac{\sqrt{x}}{12}\right)^{\frac{2}{3}}\) *	A1*	2.1
	(3)

Notes

M1: Sets \(12x^2 + x - 16\sqrt{x} = 0\) and divides by \(\sqrt{x}\) or equivalent e.g. divides by \(x\) and multiplies by \(\sqrt{x}\)

dM1: Makes the term in \(x^{\frac{3}{2}}\) the subject of the formula

A1*: A correct and rigorous argument leading to the given solution.

Alternative - working backwards:
\(x = \left(\dfrac{4}{3} - \dfrac{\sqrt{x}}{12}\right)^{\frac{2}{3}} \Rightarrow x^{\frac{3}{2}} = \dfrac{4}{3} - \dfrac{\sqrt{x}}{12} \Rightarrow 12x^{\frac{3}{2}} = 16 - \sqrt{x} \Rightarrow 12x^2 = 16\sqrt{x} - x \Rightarrow 12x^2 - 16\sqrt{x} + x = 0\)

M1: For raising to power of 3/2 both sides. dM1: Multiplies through by \(\sqrt{x}\). A1: Achieves printed answer and makes a minimal comment e.g. tick, #, QED, true etc.

Mark scheme (c)

Scheme	Marks	AO
\(x_2 = \sqrt[3]{\left(\dfrac{4}{3} - \dfrac{\sqrt{2}}{12}\right)^2}\)	M1	1.1b
\(x_2 =\) awrt 1.13894	A1	1.1b
\(x = 1.15650\)	A1	2.2a
	(3)
	(10 marks)

Notes

M1: Attempts to use the iterative formula with \(x_1 = 2\). This is implied by sight of \(x_2 = \left(\dfrac{4}{3} - \dfrac{\sqrt{2}}{12}\right)^{\frac{2}{3}}\) or awrt 1.14

A1: \(x_2 =\) awrt 1.13894

A1: Deduces that \(x = 1.15650\)