0606.14.7: Apply differentiation to connected rates of change, small increments and approximations: past paper questions

Real questions on 0606.14.7: apply differentiation to connected rates of change, small increments and approximations

The newest 14 of 14 written questions. Open one to see it on the real paper.

1. 0606/12 · May/June 2026 · Q8[5]
   8 Variables $x$ and $y$ are related by the equation $y=\frac{5x+1}{x-2}$. Using calculus, find the approximate change in $x$ when $y$ increases from 4 by the small amount $h$.See it on the paper →
2. 0606/12 · October/November 2025 · Q12b[2]
   12 A curve has equation $y=\frac{{\mathrm{e}}^{(x^2)}}{x-2}$ for $x<2$. (b) Find the corresponding rate of change of $x$.See it on the paper →
3. 0606/13 · October/November 2025 · Q4b[2]
   [Diagram: A velocity-time graph with vertical axis $v{\text{ m s}}^{-1}$ and horizontal axis $t\text{ s}$. The graph starts at the origin $(0,0)$, rises linearly to a point at $t=T$, remains horizontal until $t=4$, and then decreases linearly to the horizontal axis at $t=8$. Dashed vertical lines are shown at $t=T$ and...See it on the paper →
4. 0606/21 · October/November 2025 · Q8b[3]
   In this question, the units are metres and seconds. A particle $P$ is travelling in a straight line through a fixed point $O$. At time $t$ its acceleration, $a$, is given by $a=(2t-3{)}^2$, where $t⩾0$. When $t=3$, $P$ has a velocity of 6. (b) Use calculus to find the approximate change in $v$ when $t$ incr...See it on the paper →
5. 0606/21 · May/June 2025 · Q5[4]
   5 Given that $y=x^2\tan \frac{x}{2}$, use calculus to find the approximate change in $y$ as $x$ increases from $\frac{\pi }{3}$ to $\frac{\pi }{3}+h$, where $h$ is small.See it on the paper →
6. 0606/21 · May/June 2025 · Q9d(ii)[5]
   In this question, all lengths are in centimetres and all angles are in radians. [Diagram: A sector AOB with centre O, radius 15, and angle $\frac{6\pi }{5}$. A cone formed from the sector with slant height 15 and points A and B touching at the rim.] The diagram shows a sector $AOB$ of a circle, centre $O$, radius 15. An...See it on the paper →
7. 0606/22 · February/March 2025 · Q6c[3]
   It is given that $y=\frac{\ln (2x^2+1)}{x+2}$. (c) Given that $y$ is decreasing by $0.4$ units per second, find the corresponding rate of change in $x$ when $x=2.$See it on the paper →
8. 0606/23 · October/November 2024 · Q12a[2]
   12 A particle moves in a straight line. Its velocity, $v{\text{ ms}}^{-1}$, at time $t$ seconds is given by $$v=\cos t-\sin t.\text{\hspace{1em}(1)}$$ (a) Find the acceleration, $a{\text{ ms}}^{-2}$, when $t=\frac{\pi }{3}.$See it on the paper →
9. 0606/21 · May/June 2023 · Q5[4]
   The volume, $V$, of a sphere of radius $r$ is given by $V=\frac{4}{3}\pi r^3$. The volume of a sphere is increasing at a constant rate of $24{\text{ cm}}^3{\text{ s}}^{-1}$. Find the rate of increase of the radius when the radius is $6\text{ cm}$.See it on the paper →
10. 0606/21 · May/June 2022 · Q9b[4]

    4-mark “Find” question — its maths and diagrams are best seen on the paper.

    See it on the paper →
11. 0606/12 · February/March 2022 · Q5b[2]
    Variables $x$ and $y$ are such that $y=\frac{\ln (2x^2-3)}{3x}$. (b) Hence find the approximate change in $y$ when $x$ increases from $2$ to $2+h$, where $h$ is small.See it on the paper →
12. 0606/12 · February/March 2022 · Q5c[2]
    Variables $x$ and $y$ are such that $y=\frac{\ln (2x^2-3)}{3x}$. (c) At the instant when $x=2$, $y$ is increasing at the rate of 4 units per second. Find the corresponding rate of increase in $x$.See it on the paper →
13. 0606/23 · May/June 2021 · Q4[4]
    Variables $x$ and $y$ are such that $y=\frac{\sin x}{\cos x}$. Using differentiation, find the approximate change in $y$ as $x$ increases from $-\frac{\pi }{4}$ to $h-\frac{\pi }{4}$, where $h$ is small.See it on the paper →
14. 0606/11 · May/June 2020 · Q3[4]
    The radius, $r$ cm, of a circle is increasing at the rate of $5{\text{ cm s}}^{-1}$. Find, in terms of $\pi$, the rate at which the area of the circle is increasing when $r=3$.See it on the paper →

What examiners said

Only a small minority of candidates gave fully correct solutions for this question. The majority of candidates who gave partially correct solutions provided a relevant chain rule. Although many candidates recognised this was a rate of change question many were working with an incorrect expression for r and so made little progress. Candidates often muddled their notation and did not always realise...

— Examiner report, May/June 2025, 0606/21 Q9(d)(ii) (Cambridge)

The most common error made by candidates was not appreciating that y was decreasing in value. A correct rate of change approach was made by most candidates, but too many gave an answer of 4.71 rather than – 4.71.

— Examiner report, February/March 2025, 0606/22 Q6(c) (Cambridge)

This question was well answered. Most candidates recognised that this was a rate of change question requiring the use of the chain rule and applied it clearly and accurately. Poor notation using inappropriate variables occasionally led to errors. The accuracy of the final answer was sometimes insufficient with answers not given to 3 significant figures, as stated in the general rubric. Candidates...

— Examiner report, May/June 2023, 0606/21 Q5 (Cambridge)