18M.1.SL.TZ2.T_14

pestleMathematicsAISLPaper 118M· sl-5-1-introduction-of-differential-calculussource ↗

Consider the function $f\left( x \right) = \frac{{{x^4}}}{4}$ .

Find f'(x)

[1]

Find the gradient of the graph of f at $x = - \frac{1}{2}$ .

[2]

Find the x-coordinate of the point at which the normal to the graph of f has gradient ${ - \frac{1}{8}}$ .

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

x³ (A1) (C1)

Note: Award (A0) for $\frac{{4{x^3}}}{4}$ and not simplified to x³.

[1 mark]

${\left( { - \frac{1}{2}} \right)^3}$ (M1)

Note: Award (M1) for correct substitution of ${ - \frac{1}{2}}$ into their derivative.

${ - \frac{1}{8}}$ (−0.125) (A1)(ft) (C2)

Note: Follow through from their part (a).

[2 marks]

x³ = 8 (A1)(M1)

Note: Award (A1) for 8 seen maybe seen as part of an equation y = 8x + c, (M1) for equating their derivative to 8.

(x =) 2 (A1) (C3)

Note: Do not accept (2, 4).

[3 marks]

Examiners’ report

[N/A]