Edexcel A-Level Mathematics A2 P4.5 Differentiation Questions

Practise differentiating implicit and parametric curves, forming tangent or normal equations, and applying connected rates in geometric models.

Exam points

use dy/dx from parametric or implicit curves to form tangent and normal equations
apply the chain rule to linked rates in volumes, areas, radii and lengths
work with diagrams or given formulae to express one variable before differentiating

Covered syllabus points

P4.5.1 - Differentiation of simple functions P4.5.2 - Formation of simple differential

[Maximum number: 7]

Figure 4

Figure 4 shows a sketch of the curve C with parametric equations

x=\sec t \quad y=\sqrt{3} \tan \left(t+\frac{\pi}{3}\right) \quad \frac{\pi}{6}<t<\frac{\pi}{2}

(a)

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of t

[ 3 ]

$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sec t \tan t$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=\sqrt{3} \sec ^{2}\left(t+\frac{\pi}{3}\right)$ $\left(\frac{\mathrm{d} y}{\mathrm{~d} x}=\right) \frac{\mathrm{d} y}{\mathrm{~d} t} \div \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{\sqrt{3} \sec ^{2}\left(t+\frac{\pi}{3}\right)}{\sec t \tan t}$ oe
(3)

(a) Alt

$x=\sec t \quad y=\sqrt{3} \frac{\tan t+\sqrt{3}}{1-\sqrt{3} \tan t}$ $\frac{\mathrm{d} x}{\mathrm{~d} t}=\sec t \tan t$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=\sqrt{3} \frac{(1-\sqrt{3} \tan t) \sec ^{2} t-(\tan t+\sqrt{3})\left(-\sqrt{3} \sec ^{2} t\right)}{(1-\sqrt{3} \tan t)^{2}}($ oe )
$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} \div \frac{\mathrm{d} x}{\mathrm{~d} t}=\sqrt{3} \frac{(1-\sqrt{3} \tan t) \sec ^{2} t-(\tan t+\sqrt{3})\left(-\sqrt{3} \sec ^{2} t\right)}{\sec t \tan t(1-\sqrt{3} \tan t)^{2}}$ oe

(b)

Find an equation for the tangent to C at the point where $t=\frac{\pi}{3}$

Give your answer in the form y=m x+c, where m and c are constants.

[ 4 ]

$t=\frac{\pi}{3}\Rightarrow x=2,\quad y=-3$ $\left.\frac{dy}{dx}\right|_{t=\pi/3} =\frac{\sqrt3\sec^2(2\pi/3)}{\sec(\pi/3)\tan(\pi/3)} =2$ y+3=2(x-2)y=2x-7

[Maximum number: 4]

Figure 2

A cone, shown in Figure 2, has
- fixed height 5 cm
- base radius $r \mathrm{~cm}$
- slant height $l \mathrm{~cm}$

Given that the base radius is increasing at a constant rate of 3 cm per minute,

find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm. Give your answer in $\mathrm{cm}^{2}$ per minute to one decimal place.
[The total surface area, S, of a cone is given by the formula $S=\pi r^{2}+\pi r l$ ]

$S=\pi r^2+\pi r\sqrt{25+r^2}$ $\frac{dS}{dr}=2\pi r+\pi\sqrt{25+r^2}+\frac{\pi r^2}{\sqrt{25+r^2}}$ $\frac{dS}{dt}=\frac{dS}{dr}\frac{dr}{dt}$ $\frac{dS}{dt} =\left(2\pi(1.5)+\pi\sqrt{27.25}+\frac{\pi(1.5)^2}{\sqrt{27.25}}\right)3 =81.5\text{ cm}^2\text{/min}$

[Maximum number: 7]

Figure 3

Figure 3 shows a sketch of the curve C with parametric equations

x=6 t-3 \sin 2 t \quad y=2 \cos t \quad 0 \leqslant t \leqslant \frac{\pi}{2}

The curve meets the y-axis at 2 and the x-axis at k, where k is a constant.

(a)

Use parametric differentiation to show that

\frac{\mathrm{d} y}{\mathrm{~d} x}=\lambda \operatorname{cosec} t

where $\lambda$ is a constant to be found.

The point P with parameter $t=\frac{\pi}{4}$ lies on C.
The tangent to C at the point P cuts the y-axis at the point N.

[ 4 ]

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{-2 \sin t}{6-6 \cos 2 t}$

M1 A1

$=\frac{-2 \sin t}{6-6\left(1-2 \sin ^{2} t\right)}=\frac{-2 \sin t}{12 \sin ^{2} t}=-\frac{1}{6} \operatorname{cosec} t$

M1 A1*

(4)

(b)

Find the exact y coordinate of N, giving your answer in simplest form.

The region bounded by the curve, the x-axis and the y-axis is rotated through $2 \pi$ radians about the x-axis to form a solid of revolution.

[ 3 ]

$P=\left(\frac{3\pi}{2}-3,\sqrt2\right)$
For example,
$y-\sqrt2=-\frac{\sqrt2}{6}\left(0-\left(\frac{3\pi}{2}-3\right)\right)$
or
$\sqrt2=-\frac{\sqrt2}{6}\left(\frac{3\pi}{2}-3\right)+c$ $y=\frac{\pi\sqrt2}{4}+\frac{\sqrt2}{2}$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)