June 2018 Paper 1 Q10

Notes

Note: On e pen this is set up as B1 M1 M1 A1. We are scoring it B1 M1 A1 A1

B1: Gives the correct fraction for the gradient of the chord either \(\dfrac{(x + h)^3 - x^3}{h}\) or \(\dfrac{(x + \delta x)^3 - x^3}{\delta x}\)
It may also be awarded for \(\dfrac{(x + h)^3 - x^3}{x + h - x}\) oe. It may be seen in an expanded form
It does not have to be linked to the gradient of the chord

M1: Attempts to expand \((x + h)^3\) or \((x + \delta x)^3\) Look for two correct terms, most likely \(x^3 + \ldots + h^3\)
This is independent of the B1

A1: Achieves gradient (of chord) is \(3x^2 + 3xh + h^2\) or exact un simplified equivalent such as \(3x^2 + 2xh + xh + h^2\). Again, there is no requirement to state that this expression is the gradient of the chord

A1*: CSO. Requires correct algebra and making a link between the gradient of the chord and the gradient of the curve. See below how the link can be made. The words "gradient of the chord" do not need to be mentioned but derivative, \(\mathrm{f}'(x)\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\), \(y'\) should be. Condone invisible brackets for the expansion of \((x + h)^3\) as long as it is only seen at the side as intermediate working.
Requires either

• \(\mathrm{f}'(x) = \lim_{h \to 0} \dfrac{(x + h)^3 - x^3}{h} = 3x^2 + 3xh + h^2 = 3x^2\)
• Gradient of chord \(= 3x^2 + 3xh + h^2\) As \(h \to 0\) Gradient of chord tends to the gradient of curve so derivative is \(3x^2\)
• \(\mathrm{f}'(x) = \lim_{h \to 0} 3x^2 + 3xh + h^2 = 3x^2\)
• Gradient of chord \(= 3x^2 + 3xh + h^2\) when \(h \to 0\) gradient of curve \(= 3x^2\)
• Do not allow \(h = 0\) alone without limit being considered somewhere:
  so don’t accept \(h = 0 \Rightarrow \mathrm{f}'(x) = 3x^2 + 3x \times 0 + 0^2 = 3x^2\)

Alternative: B1: Considers \(\dfrac{(x + h)^3 - (x - h)^3}{2h}\)   M1: As above   A1: \(\dfrac{6x^2h + 2h^3}{2h} = 3x^2 + h^2\)
(corrected from the printed mark scheme: “\(6x^2h^2 + 2h^3\)”; \((x + h)^3 - (x - h)^3 = 6x^2h + 2h^3\))