Edexcel A-Level Mathematics AS P2.7 Differentiation Questions

Practise applying differentiation to curves and practical models, finding stationary points, ranges of increase or decrease, and justified extrema.

Exam points

set dy/dx=0 to find stationary coordinates or unknown values in a model
use d²y/dx² or sign changes to justify a maximum or minimum
solve derivative inequalities to state where a function is increasing or decreasing

Covered syllabus points

P2.7.1 - Applications of differentiation

[Maximum number: 3]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Figure 3

Figure 3 shows
- the curve $C_{1}$ with equation $y=x^{3}-5 x^{2}+3 x+14$
- the circle $C_{2}$ with centre T

The point T is the minimum turning point of $C_{1}$
Using Figure 3 and calculus,

find the coordinates of T

The curve $C_{1}$ intersects the circle $C_{2}$ at the point A with x coordinate 2

$y=x^{3}-5 x^{2}+3 x+14 \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}=3 x^{2}-10 x+3=0$
Roots are $3, \frac{1}{3} \Rightarrow$ when $x=3, y=" 3^{\prime \prime 3}-5 \times " 3^{\prime \prime 2}+3 \times " 3 "+14=\ldots$ .
Centre is (3,5)
(3)

[Maximum number: 3]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

f(x)=4 x^{3}+13 x^{2}-10 x+8

Find the range of values of x for which f(x) is decreasing.

$f'(x)=12x^2+26x-10$ $12x^2+26x-10=0$ $2(6x^2+13x-5)=0$ (2x+5)(3x-1)=0 $x=-\frac52,\ \frac13$ $f(x)\text{ is decreasing for }-\frac52<x<\frac13$

[Maximum number: 6]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 2

Figure 2 shows the plan view of the design for a stage at a trade fair.
The shape of the stage ABCDEFA, consists of a rectangle ACDF joined to two congruent sectors of circles. ABC is a sector of a circle centre A and FDE is a sector of a circle centre F.

Given that A C=2 r metres, C D=r metres, angle $D F E=\theta$ radians and the area of the stage is $30 \mathrm{~m}^{2}$ ,

(a)

Use calculus to find the minimum value for P, giving your answer in the form $a \sqrt{b}$ , where a and b are integers to be found.

[ 4 ]

$\frac{dP}{dr}=4-\frac{30}{r^2}$ $\frac{dP}{dr}=0\Rightarrow r=\sqrt{\frac{15}{2}}$ $P=4\sqrt{\frac{15}{2}}+\frac{30}{\sqrt{\frac{15}{2}}}=4\sqrt{30}$

(b)

Justify that the value of P found in part (b) is the minimum.

[ 2 ]

$\frac{d^2P}{dr^2}=\frac{60}{r^3}$
At $r=\sqrt{\frac{15}{2}}>0$ ,
$\frac{d^2P}{dr^2}>0$
Hence the value of P is a minimum.

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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)