2022 H2 Mathematics Paper 1 Question 7

Differentiation II: Maxima, Minima, Rates of Change

Answers

(a)

(e13,13e).{\left( \mathrm{e}^{\frac{1}{3}}, \frac{1}{3\mathrm{e}} \right).}

(b)

29−118ln⁡3.{\frac{2}{9} - \frac{1}{18} \ln 3.}

Full solutions

(a)

dydx=−3x−4ln⁡x+x−3(1x)=−3ln⁡xx4+1x4=1−3ln⁡xx4  ■\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= -3x^{-4} \ln x + x^{-3} \left(\frac{1}{x}\right) \\ &= \frac{-3 \ln x}{x^4} + \frac{1}{x^4} \\ &= \frac{1-3\ln x}{x^4} \; \blacksquare \end{align*}
At turning point,
dydx=01−3ln⁡xx4=01−3ln⁡x=0ln⁡x=13x=e13y=(e13)−3ln⁡e13=e−1⋅13=13e\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= 0 \\ \frac{1-3\ln x}{x^4} &= 0 \\ 1 - 3 \ln x &= 0 \\ \ln x &= \frac{1}{3} \\ x &= \mathrm{e}^{\frac{1}{3}} \\ y &= \left( \mathrm{e}^{\frac{1}{3}} \right)^{-3} \ln \mathrm{e}^{\frac{1}{3}} \\ &= \mathrm{e}^{-1} \cdot \frac{1}{3} \\ &= \frac{1}{3\mathrm{e}} \end{align*}
Hence the coordinates of the turning point of C{C} is (e13,13e).  ■{\left( \mathrm{e}^{\frac{1}{3}}, \frac{1}{3\mathrm{e}} \right). \; \blacksquare}

(b)

Area required=∫13x−3ln⁡xdx=[x−2−2ln⁡x]13−∫13x−2−2⋅1xdx=−ln⁡318+0+∫13x−32dx=−ln⁡318+[x−2−4]13=−ln⁡318−136+14=(29−ln⁡318) units2  ■\begin{align*} & \textrm{Area required} \\ &= \int_1^3 x^{-3} \ln x \mathop{\mathrm{d}x} \\ &= \Bigl[ \frac{x^{-2}}{-2} \ln x \Bigr]_1^3 - \int_1^3 \frac{x^{-2}}{-2} \cdot \frac{1}{x} \mathop{\mathrm{d}x} \\ &= -\frac{\ln 3}{18}+0 + \int_1^3 \frac{x^{-3}}{2} \mathop{\mathrm{d}x} \\ &= -\frac{\ln 3}{18} + \Bigl[ \frac{x^{-2}}{-4} \Bigr]_1^3 \\ &= -\frac{\ln 3}{18} - \frac{1}{36} + \frac{1}{4} \\ &= \left( \frac{2}{9} - \frac{\ln 3}{18} \right) \textrm{ units}^2 \; \blacksquare \end{align*}

Question Commentary

This question is a relative straightforward calculus question (the 2022 papers seem to have relatively more on this topic than in the past). Part (a) is about differentiation via product rule and finding turning points.

Meanwhile, part (b) is about finding area using integration. Because we need an exact answer, we cannot rely on our graphing calculator and will need to apply integration by parts.