CIE Application of derivative (2020 Additional)

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1. (CIE $0606 / 2018 / \mathrm{w} / 11 / \mathrm{q} 6)$

A curve has equation $y=\frac{\ln \left(2 x^{2}+3\right)}{5 x+2}$

(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{5}{4} \ln 3$ when $x=0$.

(ii) Hence find the equation of the tangent to the curve at the point where $x=0$.

2. $\left(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 11 / \mathrm{q}^{9}\right)$

Variables $s$ and $t$ are such that $s=4 t+3 \mathrm{e}^{-t}$.

(i) Find the value of $s$ when $t=0$.

(ii) Find the exact value of $t$ when $\frac{d s}{d t}=2$.

(iii) Find the approximate increase in $s$ when $t$ increases from $\ln 5$ to $\ln 5+h$, where $h$ is small.

3. (CIE $0606 / 2018 / \mathrm{w} / 12 / \mathrm{q} 4)$

$$y=x^{3} \ln (2 x+1)$$

(i) Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=0.3$. You must show all your working.

(ii) Hence find the approximate increase in $y$ when $x$ increases from $0.3$ to $0.3+h$, where $h$ is small.

4. (CIE $0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 4)$

In this question, the units of $x$ are radians and the units of $y$ are centimetres.

It is given that $y=(1+\cos 3 x)^{16}$.

(i) Find the value of $\frac{d y}{d x}$ when $x=\frac{\pi}{2}$.

Given also that $y$ is increasing at a rate of $6 \mathrm{~cm} \mathrm{~s}^{-1}$ when $x=\frac{\pi}{2}$,

(ii) find the corresponding rate of change of $x$. [2]

5. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 9)$

Find the equation of the normal to the curve $y=\frac{\ln \left(3 x^{2}+1\right)}{x^{2}}$ at the point where $x=2$, giving your answer in the form $y=m x+c$, where $m$ and $c$ are correct to 2 decimal places. You must show all your working. $[8]$

6. (CIE $0606 / 2018 / \mathrm{w} / 21 / \mathrm{q} 9)$

In this question, all lengths are in metres.

The diagram shows a window formed by a semi-circle of radius $r$ on top of a rectangle with dimensions $2 r$ by $y$. The total perimeter of the window is 5 .

(i) Find $y$ in terms of $r$.

(ii) Show that the total area of the window is $A=5 r-\frac{\pi r^{2}}{2}-2 r^{2}$.

(iii) Given that $r$ can vary, find the value of $r$ which gives a maximum area of the window and find this area. (You are not required to show that this area is a maximum.)

7. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 22 / \mathrm{q} 3)$

A curve has equation $y=\frac{x^{3}}{\sin 2 x}$. Find

(i) $\frac{\mathrm{d} y}{\mathrm{~d} x}$,

(ii) the equation of the tangent to the curve at the point where $x=\frac{\pi}{4}$. [3]

8. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 23 / \mathrm{q} 10)$

The equation of a curve is $y=x^{2} \sqrt{3+x}$ for $x \geqslant-3$

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Find the equation of the tangent to the curve $y=x^{2} \sqrt{3+x}$ at the point where $x=1$.

(iii) Find the coordinates of the turning points of the curve $y=x^{2} \sqrt{3+x}$.

9. (CIE $0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 6)$

Find the equation of the normal to the curve $y=\sqrt{8 x+5}$ at the point where $x=\frac{1}{2}$, giving your answer in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.

10. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 12 / \mathrm{q} 10)$

A curve is such that when $x=0$, both $y=-5$ and $\frac{\mathrm{d} y}{\mathrm{~d} x}=10$. Given that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=4 \mathrm{e}^{2 x}+3$, find

(i) the equation of the curve,

(ii) the equation of the normal to the curve at the point where $x=\frac{1}{4}$.

11. (CIE $0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 10)$

The diagram shows an open container in the shape of a cuboid of width $x \mathrm{~cm}$, length $4 x \mathrm{~cm}$ and height $h \mathrm{~cm}$. The volume of the container is $800 \mathrm{~cm}^{3}$.

(i) Show that the external surface area, $S \mathrm{~cm}^{2}$, of the open container is such that $S=4 x^{2}+\frac{2000}{x}$.

(ii) Given that $x$ can vary, find the stationary value of $S$ and determine its nature.

12. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 12 / \mathrm{q} 7)$

It is given that $y=\left(1+\mathrm{e}^{x^{2}}\right)(x+5)$

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Find the approximate change in $y$ as $x$ increases from $0.5$ to $0.5+p$, where $p$ is small.

(iii) Given that $y$ is increasing at a rate of 2 units per second when $x=0.5$, find the corresponding rate of change in $x$.

13. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 12 / \mathrm{q} 9)$

The area of a sector of a circle of radius $r \mathrm{~cm}$ is $36 \mathrm{~cm}^{2}$.

(i) Show that the perimeter, $P \mathrm{~cm}$, of the sector is such that $P=2 r+\frac{72}{r}$.

(ii) Hence, given that $r$ can vary, find the stationary value of $P$ and determine its nature.

14. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 12 / \mathrm{q} 9)$

A solid circular cylinder has a base radius of $r \mathrm{~cm}$ and a height of $h \mathrm{~cm}$. The cylinder has a volume of $1200 \pi \mathrm{cm}^{3}$ and a total surface area of $S \mathrm{~cm}^{2}$

(i) Show that $S=2 \pi r^{2}+\frac{2400 \pi}{r}$.

(ii) Given that $h$ and $r$ can vary, find the stationary value of $S$ and determine its nature.

15. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 13 / \mathrm{q} 3)$

(i) Differentiate $y=\left(3 x^{2}-1\right)^{-\frac{1}{3}}$ with respect to $x$.

(ii) Find the approximate change in $y$ as $x$ increases from $\sqrt{3}$ to $\sqrt{3}+p$, where $p$ is small.

(iii) Find the equation of the normal to the curve $y=\left(3 x^{2}-1\right)^{-\frac{1}{3}}$ at the point where $x=\sqrt{3}$.

16. (CIE 0606/2019/w/13/q9)

The diagram shows a sector $O P Q$ of the circle centre $O$, radius $3 r \mathrm{~cm}$. The points $S$ and $R$ lie on $O P$ and $O Q$ respectively such that $O R S$ is a sector of the circle centre $O$, radius $2 r \mathrm{~cm}$. The angle $P O Q=\theta$ radians. The perimeter of the shaded region $P Q R S$ is $100 \mathrm{~cm}$.

(i) Find $\theta$ in terms of $r$.

(ii) Hence show that the area, $A \mathrm{~cm}^{2}$, of the shaded region $P Q R S$ is given by $A=50 r-r^{2}$.

(iii) Given that $r$ can vary and that $A$ has a maximum value, find this value of $A$.

(iv) Given that $A$ is increasing at the rate of $3 \mathrm{~cm}^{2} \mathrm{~s}^{-1}$ when $r=10$, find the corresponding rate of change of $r$.

(v) Find the corresponding rate of change of $\theta$ when $r=10$. $[3]$

17. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 11 / \mathrm{q} 3)$

The radius, $r \mathrm{~cm}$, of a circle is increasing at the rate of $5 \mathrm{cms}^{-1}$. Find, in terms of $\pi$, the rate at which the area of the circle is increasing when $r=3$.

18. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 11 / \mathrm{q} 5)$

Find the equation of the tangent to the curve $y=\frac{\ln \left(3 x^{2}-1\right)}{x+2}$ at the point where $x=1$. Give your answer in the form $y=m x+c$, where $m$ and $c$ are constants correct to 3 decimal places.

19. $\left(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 12 / \mathrm{q}^{2}\right)$

The volume, $V$, of a sphere of radius $r$ is given by $V=\frac{4}{3} \pi r^{3}$.

The radius, $r \mathrm{~cm}$, of a sphere is increasing at the rate of $0.5 \mathrm{cms}^{-1}$. Find, in terms of $\pi$, the ratc of change of the volume of the sphere when $r=0.25$.

20. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 12 / \mathrm{q} 4)$

The tangent to the curve $y=\ln \left(3 x^{2}-4\right)-\frac{x^{3}}{6}$, at the point where $x=2$, meets the $y$-axis at the point $P$. Find the exact coordinates of $P$.

21. (CIE $0606 / 2020 / \mathrm{s} / 12 / \mathrm{q} 7)$

(a) Given that $y=\left(x^{2}-1\right) \sqrt{5 x+2}$, show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{A x^{2}+B x+C}{2 \sqrt{5 x+2}}$, where $A, B$ and $C$ are integers.

(b) Find the coordinates of the stationary point of the curve $y=\left(x^{2}-1\right) \sqrt{5 x+2}$, for $x>0 .$ Give each coordinate correct to 2 significant figures.

(c) Determine the nature of this stationary point.

22. A curve has equation $y=\frac{\ln \left(3 x^{2}-5\right)}{2 x+1}$ for $3 x^{2}>5$

(a) Find the equation of the normal to the curve at the point where $x=\sqrt{2}$.

(b) Find the approximate change in $y$ as $x$ increases from $\sqrt{2}$ to $\sqrt{2}+h$, where $h$ is small.

23. (CIE 0606/2020/w/12/q9)

A curve has equation $y=(2x-1)\sqrt{4x+3}.$

(a) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{4(A x+B)}{\sqrt{4 x+3}}$, where $A$ and $B$ are constants.

(b) Hence write down the $x$-coordinate of the stationary point of the curve.

(c) Determine the nature of this stationary point.

24. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 13 / \mathrm{q} 10)$

(a) Given that $y=x \sqrt{x+2}$, show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{A x+B}{2 \sqrt{x+2}}$, where $A$ and $B$ are constants.

(b) Find the exact coordinates of the stationary point of the curve $y=x \sqrt{x+2}$.

(c) Determine the nature of this stationary point.

25. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 21 / \mathrm{q} 10)$

The gradient of the normal to a curve at the point $(x, y)$ is given by $\frac{x}{x+1}$.

(a) Given that the curve passes through the point $(1,4)$, show that its equation is $y=5-\ln x-x$.

(b) Find, in the form $y=m x+c$, the equation of the tangent to the curve at the point where $x=3$. $[3]$

26. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 21 / \mathrm{q} 11)$

The equation of a curve is $y=x \sqrt{16-x^{2}}$ for $0 \leqslant x \leqslant 4$

(a) Find the exact coordinates of the stationary point of the curve.

(b) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(16-x^{2}\right)^{\frac{3}{2}}$ and hence evaluate the area enclosed by the curve $y=x \sqrt{16-x^{2}}$ and the lines $y=0, x=1$ and $x=3$

27. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 12)$

(a) Find the $x$-coordinates of the stationary points of the curve $y=\mathrm{e}^{3 x}(2 x+3)^{6}$.

(b) A curve has equation $y=f(x)$ and has exactly two stationary points. Given that $f^{\prime \prime}(x)=4 x-7$ $f^{\prime}(0.5)=0$ and $f^{\prime}(3)=0$, use the second derivative test to determine the nature of each of the stationary points of this curve.

(c) In this question all lengths are in centimetres.

The diagram shows a solid cuboid with height $h$ and a rectangular base measuring $4 x$ by $x$. The volume of the cuboid is $40 \mathrm{~cm}^{3}$. Given that $x$ and $h$ can vary and that the surface area of the cuboid has a minimum value, find this value. $[5]$

28. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 1)$

Variables $x$ and $y$ are such that $y=\sin x+\mathrm{e}^{-x}$. Use differentiation to find the approximate change in $y$ as $x$ increases from $\frac{\pi}{4}$ to $\frac{\pi}{4}+h$, where $h$ is small.

29. (CIE $0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 11)$

A container is a circular cylinder, open at one end, with a base radius of $r \mathrm{~cm}$ and a height of $h \mathrm{~cm}$. The volume of the container is $1000 \mathrm{~cm}^{3}$. Given that $r$ and $h$ can vary and that the total outer surface area of the container has a minimum value, find this value. $[8]$

30. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 5)$

DO NOT USE A CALCULATOR IN THIS QUESTION.

(a) Find the equation of the tangent to the curve $y=x^{3}-6 x^{2}+3 x+10$ at the point where $x=1$.

(b) Find the coordinates of the point where this tangent meets the curve again.

31. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 6)$

(a) Find the equation of the tangent to the curve $2 y=\tan 2 x+7$ at the point where $x=\frac{\pi}{8}$. Give your answer in the form $a x-y=\frac{\pi}{b}+c$, where $a, b$ and $c$ are integers.

(b) This tangent intersects the $x$-axis at $P$ and the $y$-axis at $Q$. Find the length of $P Q$.

32. (CIE $0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 9)$

Find the values of $x$ for which $12 x^{2}-20 x+5<(2 x+1)(x-1)$

33. (CIE $0606 / 2020 / \mathrm{s} / 23 / \mathrm{q} 11)$

In this question all lengths are in centimetres.

The volume, $V$, of a cone of height $h$ and base radius $r$ is given by $V=\frac{1}{3} \pi r^{2} h$

The diagram shows a large hollow cone from which a smaller cone of height 180 and base radius 90 has been removed. The remainder has been fitted with a circular base of radius 90 to form a container for water. The depth of water in the container is $w$ and the surface of the water is a circle of radius $R$.

(a) Find an expression for $R$ in terms of $w$ and show that the volume $V$ of the water in the container is given by $\quad V=\frac{\pi}{12}(w+180)^{3}-486000 \pi$

(b) Water is poured into the container at a rate of $10000 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$. Find the rate at which the depth of the water is increasing when $w=10$.

34. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 23 / \mathrm{q} 4)$

It is given that $y=\ln (1+\sin x)$ for $0<x<\pi$.

(a) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$,

(b) Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=\frac{\pi}{6}$, giving your answer in the form $\frac{1}{\sqrt{a}}$, where $a$ is an integer.

(c) Find the values of $x$ for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=\tan x$

35. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 23 / \mathrm{q} 8)$

(a) Differentiate $y=\tan (x+4)-3 \sin x$ with respect to $x$.

(b) Variables $x$ and $y$ are such that $y=\frac{\ln (2 x+5)}{2 \mathrm{e}^{3 x}}$. Use differentiation to find the approximate change in $y$ as $x$ increases from $I$ to $1+h$, where $h$ is small.

36. (CIE $0606 / 2020 / \mathrm{w} / 23 / \mathrm{q} 9)$

The rectangle $A B C D E$ represents a ploughed field where $A B=300 \mathrm{~m}$ and $A E=400 \mathrm{~m}$. Joseph needs to walk from $A$ to $D$ in the least possible time. He can walk at $0.9 \mathrm{~ms}^{-1}$ on the ploughed field and at $1.5 \mathrm{~ms}^{-1}$ on any part of the path $B C D$ along the edge of the field. He walks from $A$ to $C$ and then from $C$ to $D$. The distance $B C=x \mathrm{~m}$.

(a) Find, in terms of $x$, the total time, $T \mathrm{~s}$, Joseph takes for the journey.

(b) Given that $x$ can vary, find the value of $x$ for which $T$ is a minimum and hence find the minimum value of $T$. [6]

1. (i) Show

(ii) $y=\left(-\frac{5}{4} \ln 3\right) x+\frac{1}{2} \ln 3$

2. (i) 3

(ii) $t=\ln \frac{3}{2}$

(iii) $\delta s=\frac{17 h}{5}$

3. (i) $\frac{d y}{d x}=0.161$

(ii) $\delta y=0.161 \mathrm{~h}$

4. (i) $\frac{d y}{d x}=30$ (ii) $\frac{d x}{d t}=\frac{1}{5}$

5. $y=2.44 x-4.23$

6. (i) $y=\frac{5-2 r-\pi r}{2}$

(ii) Show

(iii) $r=0.7, A=1.75$

7. (i) $\frac{d y}{d x}=\frac{3 x^{2} \sin 2 x-2 x^{3} \cos 2 x}{(\sin 2 x)^{2}}$

(ii) $y=1.85 x-0.97$

8. (i) $\frac{d y}{d x}=\frac{x^{2}}{2}(3+x)^{\frac{1}{2}}+2 x(3+x)^{\frac{1}{2}}$

(ii) $y=\frac{17}{4} x-\frac{9}{4}$

(iii) $(0,0),(-2.4,4.46)$

9. $6 x+8 y-27=0$

10. (i) $y=e^{2 x}+\frac{3 x^{2}}{2}+8 x-6$

(ii) $y+2.26=-\frac{1}{12}\left(x-\frac{1}{4}\right)$

11. (i) Show (ii) $S_{\min }=476$

12. (i) $\frac{d y}{d x}=\left(1+e^{x^{2}}\right)+2 x e^{x^{2}}(x+5)$

(ii) $\delta y=9.35 p$ (iii) $\frac{d x}{d t}=0.214$

13. (i) Prove

(ii) $\mathrm{P}=24$

14. (i) Show (ii) $S_{\mathrm{mim}}=1340$

15. (i) $\frac{d y}{d x}=-\frac{1}{3}(6 x)\left(3 x^{2}-1\right)^{-\frac{4}{3}}$

(ii) $\delta y=-\frac{\sqrt{3}}{8} p$

(iii) $y-\frac{1}{2}=\frac{8}{\sqrt{3}}(x-\sqrt{3})$

16. (i) $\theta=\frac{100-2 r}{5 r}$

(ii) Show (iii) $A_{\max }=625$

(iv) $\frac{d r}{d t}=\frac{1}{10}$

(v) $-\frac{1}{50}$

17. $\frac{d A}{d t}=30 \pi$

18. $y=0.923 x-0.69212345678910$

19. $\frac{d V}{d t}=0.125 \pi$

20. $\left(0, \ln 8-\frac{1}{3}\right) 123456788910$

21. (a) $A=25, B=8, C=-5$

(b) $x=0.315, y=-1.70$ (c) Minimum

22. (a) $y=-\frac{4+\sqrt{2}}{12}(x-\sqrt{2})$

(b) $2.22 h$

23. (a) Prove

(b) $-\frac{1}{3}$ (c) minimum

24. (a) $\frac{d y}{d x}=\frac{3 x+4}{2 \sqrt{x+2}}$

(b) $x=-\frac{4}{3}, y=-\frac{4 \sqrt{6}}{9}$

(c) Minimum

25. (a) prof

(b) $y=-1.33 x+4.90$

26. (a) $(2 \sqrt{2}, 8)$

(b) $\frac{3}{2}\left(16-x^{2}\right)^{\frac{1}{2}}(-2 x), 13.2$

27. (a) $x=-1.5,-3.5$

(b) $x=0.5 \Rightarrow$ maximum, $x=3 \Rightarrow$ Minimum

(c) $81.4$

28. $\delta y=0.251 \mathrm{~h}$

29. $A=439$

30. (a) $y=-6 x+14$

(b) $(4,-10)$

32. $\delta y=7.14 h$

33. (a) $R=\frac{1}{2}(w+180)$

(b) $\frac{d w}{d t}=0.353$

34. (a) $\frac{d y}{d x}=\frac{\cos x}{1+\sin x}$

(b) $\frac{d y}{d x}=\frac{1}{\sqrt{3}}$

(c) $x=\frac{\pi}{6}, \frac{5 \pi}{6}$

35. (a) $\sec ^{2}(x+4)-3 \cos x$

(b) $\delta_{y}=-0.138 \mathrm{~h}$

36. (a) $T=\frac{\sqrt{300^{2}+x^{2}}}{0.9}+\frac{400-x}{1.5}$

(b) $x=225, T=533$