CIE Differentation and Application (Plus 2020)

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

It is given that $y=\left(x^{2}+1\right)(2 x-3)^{\frac{1}{2}}$

(i) Show that $\frac{d y}{d x}=\frac{P x^{2}+Q x+1}{(2 x-3)^{\textstyle\frac{1}{2}}}$, where $P$ and $Q$ are integers.

(ii) Hence find the equation of the normal to the curve $y=\left(x^{2}+1\right)(2 x-3)^{\frac{1}{2}}$ at the point where $x=2$, giving your answer in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.


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

The normal to the curve $y=(x-2)(3 x+1)^{\frac{1}{3}}$ at the point where $x=\frac{7}{3}$, meets the $y$-axis at the point $P$, Find the exact coordinates of the point $P$.


3. $(\mathrm{CIF} 0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 3)$

The number, $B$, of a certain type of bacteria at time $t$ days can be described by $B=200 \mathrm{e}^{2 t}+800 \mathrm{e}^{-2 \mathrm{r}}$

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

(ii) At the instant when $\frac{\mathrm{d} B}{\mathrm{~d} t}=1200$, show that $\mathrm{e}^{4 \mathrm{r}}-3 \mathrm{e}^{2 t}-4=0$

(iii) Using the substitution $u=\mathrm{e}^{2}$, or otherwise, solve $\mathrm{c}^{4 r}-3 \mathrm{c}^{2 x}-4=0$.


4. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 13 / \mathrm{q} 4)$

It is given that $y=\frac{\ln \left(2 x^{3}+5\right)}{x-1}$ for $x>1$

(i) Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=2$. You must show all your working.

(ii) Find the approximate change in $y$ as $x$ increases from 2 to $2+p$, where $p$ is small. $[1]$


5. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 21 / \mathrm{q} 2)$

Two variables $x$ and $y$ are such that $y=\frac{\ln x}{x^{3}} \quad$ for $x>0$

(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1-3 \ln x}{x^{4}}$.

(ii) Hence find the approximate change in $y$ as $x$ increases from e to $e+h$, where $h$ is small.


6. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 21 / \mathrm{q} 7)$

The variables $x, y$ and $u$ are such that $y=\tan u$ and $x=u^{3}+1$.

(i) State the rate of change of $y$ with respect to $u$. [1]

(ii) Hence find the rate of change of $y$ with respect to $x$, giving your answer in terms of $x$.


7. (CIE $0606 / 2019 / \mathrm{m} / 22 / \mathrm{q} 2)$

Variables $x$ and $y$ are related by the equation $y=\frac{\ln x}{\mathrm{e}^{x}}$.

(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1-x \ln x}{x \mathrm{e}^{K}}$.

(ii) Hence find the approximate change in $y$ as $x$ increases from 2 to $2+h$, where $h$ is small.


8. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 22 / \mathrm{q} 7)$

(i) Given that $y=x \sqrt{x^{2}+1}$, show that $\frac{d y}{d x}=\frac{a x^{2}+b}{\left(x^{2}+1\right)^{n}}$, where $a, b$ and $p$ are positive constants. [4]

(ii) Explain why the graph of $y=x \sqrt{x^{2}+1}$ has no stationary points.


9. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 11 / \mathrm{q} 4)$

It is given that $y=\frac{\tan 3 x}{\sin x}$.

(a) Find the exact value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=\frac{\pi}{3}$.

(b) Hence find the approximate change in $y$ as $x$ increases from $\frac{\pi}{3}$ to $\frac{\pi}{3}+h$, where $h$ is small. [1]

(c) Given that $x$ is increasing at the rate of 3 units per second, find the corresponding rate of change in $y$ when $x=\frac{\pi}{3}$, giving your answer in its simplest surd form.


10. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 4)$

It is given that $y=\frac{\ln \left(4 x^{2}+1\right)}{2 x-3}$

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

(ii) Find the approximate change in $y$ as $x$ increases from 2 to $2+h$, where $h$ is small. $[2]$


11. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 13 / \mathrm{q} 5)$

(i) Differentiate $\left(x^{2}+3\right) \ln \left(x^{2}+3\right)$ with respect to $x$. [3]

(ii) Hence find $\int x \ln \left(x^{2}+3\right) \mathrm{d} x$


12. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 21 / \mathrm{q} 8)$

The equation of a curve is given by $y=x \mathrm{e}^{-2 x}$.

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

(ii) Find the exact coordinates of the stationary point on the curve $y=x \mathrm{e}^{-2 x}$.

(iii) Find, in terms of e, the equation of the tangent to the curve $y=x e^{-2 x}$ at the point $\left(1, \frac{1}{\mathrm{e}^{2}}\right)$.

(iv) Using your answer to part (i), find $\int x e^{-2 x} \mathrm{~d} x$. [3]


13. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 22 / \mathrm{q} 1)$

Given that $y=\frac{\sin x}{\ln x^{2}}$, find an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$.


14. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 10)$

(i) Given that $y=\frac{\ln x}{x^{2}}$, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Find the coordinates of the stationary point on the curve $y=\frac{\ln x}{x^{2}}$.

(iii) Using your answer to part (i), find $\int \frac{\ln x}{x^{3}} \mathrm{~d} x$.

(iv) Hence evaluate $\int_{1}^{2} \frac{\ln x}{x^{3}} \mathrm{~d} x$


15. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 2)$

Given that $y=2 \sin 3 x+\cos 3 x$, show that $\frac{\mathrm{d}^{2} y}{d x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+3 y=k \sin 3 x$, where $k$ is a constant to be determined.  $[5]$


16. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 22 / \mathrm{q} 4)$

A circle has diameter $x$ which is increasing at a constant rate of $0.01 \mathrm{cms}^{-1}$. Find the exact rate of change of the area of the circle when $x=6 \mathrm{~cm}$.


17. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 5)$

At the point where $x=1$ on the curve $y=\frac{k}{(x+1)^{2}}$, the normal has a gradient of $\frac{1}{3}$.

(i) Find the value of the constant $k$. [4]

(ii) Using your value of $k$, find the equation of the tangent to the curve at $x=2$. $[3]$


18. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 2)$

Differentiate $\tan 3 x \cos \frac{x}{2}$ with respect to $x$.


19. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 6)$

A curve has equation $y=(3 x-5)^{3}-2 x$

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.

(ii) Find the exact value of the $x$-coordinate of each of the stationary points of the curve. [2]

(iii) Use the second derivative test to determine the nature of each of the stationary points. $[2]$


20. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 13 / \mathrm{q} 2)$

(a) Given that $y=\frac{e^{2 x-3}}{x^{2}+1}$, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(b) Hence, given that $y$ is increasing at the rate of 2 units per second, find the exact rate of change of $x$ when $x=2$ [3]


21. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 21 / \mathrm{q} 4)$

It is given that $y=\ln (\sin x+3 \cos x)$ for $0<x<\frac{\pi}{2}$

(a) Find $\frac{\mathrm{d} y}{\mathrm{dx}}$.

(b) Find the value of $x$ for which $\frac{d y}{d x}=-\frac{1}{2}$.


22. (CIE $0606 / 2020 / \mathrm{s} / 23 / \mathrm{q} 12)$

(a) (i) Given that $f(x)=\frac{1}{\cos x^{\prime}}$, show that $f^{\prime}(x)=\tan x \sec x$.

(ii) Hence find $\int\left(3 \tan x \sec x-\sqrt[4]{\mathrm{e}^{3 x}}\right) \mathrm{d} x$


23. (CIE 0606/2020/w/23/q7)

A curve has equation $y=x \cos x$

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

(b) Find the equation of the normal to the curve at the point where $x=\pi$, giving your answer in the form $y=m x+c$.

(c) Using your answer to part (a), find the exact value of $\int_{0}^{\frac{\pi}{6}} x \sin x \mathrm{~d} x$. $[5]$



Answer

1. (i) $\frac{d y}{d x}=\frac{5 x^{2}-6 x+1}{(2 x-3)^{\frac{1}{2}}}$

(ii) $x+9 y-47=0$

2. $\quad P=\left(0, \frac{73}{39}\right)$

3. (i) $B=1000$,

(ii) Show (iii) $t=\ln 2$

4. (i) $\frac{8}{7}-\ln 21$ (ii) $-1.90 p$

5. (i) Show (ii) $\delta y=-0.0366 h$

6. (i) $\sec ^{2} u$

(ii) $\frac{d y}{d x}=\frac{\sec ^{2}(\sqrt[3]{x-1})}{3(\sqrt[3]{x-1})^{2}}$

7. (ii) $-0.0261 h$

8. (i) $a=2, b=1, p=.5$

9. (a) $2 \sqrt{3}$

(b) $2 \sqrt{3} h$

(c) $6 \sqrt{3}$

10. (i) $\frac{d y}{d x}=\frac{(2 x-3) \frac{8 x}{\left(4 x^{2}+1\right)}-2 \ln \left(4 x^{2}+1\right)}{(2 x-3)^{2}}$

(ii) $\delta y=-4.73 h$

11. (i) $\left(x^{2}+3\right) \frac{2 x}{\left(x^{2}+3\right)}+2 x \ln \left(x^{2}+3\right)$

(ii) $\frac{1}{2}\left(x^{2}+3\right) \ln \left(x^{2}+3\right)-\frac{x^{2}}{2}(+c)$

12. (i) $\frac{d y}{d x}=e^{-2 x}(1-2 x)$

(ii) $\left(\frac{1}{2}, \frac{1}{2 e}\right)$

(iii) $y=-\frac{1}{e^{2}} x+\frac{2}{e^{2}}$ (iv) $\frac{-x e^{-2 x}}{2}-\frac{e^{-2 x}}{4}+c$

13. $\frac{d y}{d x}=\frac{\left(\ln x^{2}\right) \cos x-(\sin x)\left(\frac{2}{x}\right)}{\left(\ln x^{2}\right)^{2}}$

14. (i) $\frac{d y}{d x}=\frac{x-2 x \ln x}{x^{4}}$

(ii) $x=\sqrt{e}, y=\frac{1}{2 e}$

(iii) $\frac{1}{4 x^{2}}-\frac{\ln x}{\ln 2^{2}}(+c)$

(iv) $\frac{3}{16}-\frac{\ln 2}{8}$

15. Show

16. $\frac{d A}{d t}=0.03 \pi$

17. (i) $k=12$

(ii) $y=-\frac{8}{9} x+\frac{28}{9}$

18. $\left(3 \sec ^{2} 3 x\right) \cos \left(\frac{x}{2}\right)+\tan 3 x\left(-\frac{1}{2} \sin \frac{x}{2}\right)$

19. (i) $9(3 x-5)^{2}-2,54(3 x-5)$

(ii) $\frac{5}{3} \pm \frac{\sqrt{2}}{9}$

(iii) $\frac{5}{8}+\frac{\sqrt{2}}{9} \rightarrow \min , \frac{5}{8}-\frac{\sqrt{2}}{9} \rightarrow \max$

20. (a) $\frac{d y}{d x}=\frac{\left(x^{2}+1\right) 2 e^{2 x-3}-2 x e^{2 x-3}}{\left(x^{2}+1\right)^{2}}$

(b) $\frac{25}{3 e}$

21. (a) $\frac{d y}{d x}=\frac{\cos x-3 \sin x}{\sin x+3 \cos x}$

(b) $x=\frac{\pi}{4}$

22. (a)(i) Show

(ii) $\frac{3}{\cos x}-\frac{4}{3} e^{\frac{3 x}{4}}+c$

(b) $p=10$

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

(b) $y=x-2 \pi$

(c) $\frac{1}{2}-\frac{\pi \sqrt{3}}{12}$

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