A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 10: Numerical methods · Lesson 1

Locating roots

114 marks

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Quick questions to start.

A1
Needed today
Given $f (x) = x^{3} - 5 x + 1$ , work out $f (2)$ and $f (2.5)$ .
[2 marks]
Answer
A2
Needed today
Find the coordinates of the turning points of $y = x^{3} - 3 x + 1$ .
[3 marks]
Answer
A3
Common slip
Given $g (x) = 2 sin x - x$ , work out $g (1.8)$ and $g (1.9)$ to 3 significant figures. Use radians.
[2 marks]
Answer
A4
Needed today
Find the exact $x$ -coordinate of the stationary point of $y = e^{x} - 4 x$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Show that $x^{3} - 2 x - 5 = 0$ has a root in the interval $[2, 3]$ .

Hint Work out

f (2)

and

f (3)

. Then write all three parts of the conclusion: sign change, continuous, root.

B1
Show that $ln x + x - 4 = 0$ has a root in the interval $[2.9, 3]$ .
[3 marks]

Example 2

The equation $x^{3} - 2 x - 5 = 0$ has a root $α$ . Show that $α = 2.09$ to 2 decimal places.

Hint Every number that rounds to

2.09

lies between

2.085

and

2.095

. Test those two values.

B2
The equation $ln x + x - 4 = 0$ has a root $β$ . Show that $β = 2.926$ to 3 decimal places.
[4 marks]

Example 3

Given $f (x) = \frac{3}{x - 2} + x$ , show that $f (1) < 0$ and $f (3) > 0$ . Explain why $f (x) = 0$ has no root in $[1, 3]$ .

Hint Where is

f

not defined? A change of sign only proves a root when the function is continuous.

B3
Given $g (x) = \frac{2}{x - 1} + x$ , show that $g (0)$ and $g (2)$ have different signs. Explain why this does not prove a root in $[0, 2]$ , and show there is no root.
[5 marks]

Example 4

Given $f (x) = x^{3} - 3 x + 2$ , show that $f (0.5)$ and $f (1.5)$ are both positive, but that $f (x) = 0$ has a root in $[0.5, 1.5]$ .

Hint Factorise. A repeated root touches the axis, so the sign does not change there.

B4
Given $h (x) = x^{3} - 12 x + 16$ , show that $h (1.5)$ and $h (2.5)$ are both positive. Show that $h (x) = 0$ has a root in $[1.5, 2.5]$ all the same.
[4 marks]

Example 5

Show that $e^{x} + x^{2} - 4 = 0$ has exactly one positive root, and that it lies in $[1, 1.1]$ .

Hint For

x > 0

the function is increasing. Then use a sign change on

[1, 1.1]

B5
Show that $ln x + x^{2} - 3 = 0$ has exactly one root, and that it lies in $[1.5, 1.6]$ .
[5 marks]

Practice

A-Level

C1
Ola tests $f (x) = 3 sin x - x$ at $x = 2.2$ and $x = 2.3$ and gets $- 2.08$ and $- 2.18$ . She says there is no root in $[2.2, 2.3]$ . Find her mistake and show that there is a root.
[4 marks]

C2
The root $α$ of $3 sin x - x = 0$ near $2.3$ is to be given to 2 decimal places. Sam tests $x = 2.27$ and $x = 2.28$ and finds a sign change. Explain why this does not tell him $α$ to 2 decimal places, then find it.
[5 marks]
Answer
C3
Given $f (x) = x^{3} - 6 x^{2} + 9 x - 1$ , show that $f (0)$ and $f (4)$ have different signs. Then show that $f (x) = 0$ has three roots in $[0, 4]$ and give an interval of width $1$ for each.
[5 marks]
Answer
C4
Given $f (x) = tan x - x$ , show that $f (1.5) > 0$ and $f (1.6) < 0$ . Explain why there is no root in $[1.5, 1.6]$ .
[4 marks]
C5
Given $f (x) = 4 x^{3} - 12 x^{2} + 9 x$ , Kai works out $f (1) = 1$ and $f (2) = 2$ and says that $f (x) = 0$ has no root in $[1, 2]$ . Show that he is wrong.
[4 marks]

Exam-style questions

Edexcel exam style

D1
$f (x) = ln (2 x) - x^{2} + 3$ , $x > 0$ .
(a)
Find the exact $x$ -coordinate of the stationary point of $y = f (x)$ .
[3 marks]
Answer
(b)
Show that $f (x) = 0$ has exactly two roots.
[3 marks]
(c)
The larger root is $α$ . Show that $α$ lies in $[2.1, 2.2]$ .
[2 marks]
(d)
Show that $α = 2.11$ to 2 decimal places.
[3 marks]
Total for D1: 11 marks

D2
Leo writes: " $f (1.73) = - 0.0923$ and $f (1.74) = 0.00802$ , so the root of $x^{3} + x - 7 = 0$ is $1.74$ to 2 decimal places."
The root is 1.74 to 2 decimal places, but Leo's working does not prove it.
(a)
Explain why Leo's working is not enough.
[2 marks]
(b)
Show that the root is $1.74$ to 2 decimal places.
[3 marks]
(c)
Show that this is the only root of $x^{3} + x - 7 = 0$ .
[2 marks]
Total for D2: 7 marks
D3
The curves $y = e^{x}$ and $y = 3 x^{2}$ meet at a point where $x = β$ and $3.7 < β < 3.8$ . Show that $β$ lies in this interval and find $β$ to 2 decimal places, showing that your answer is correct to this accuracy.
[7 marks]
Answer
D4
$f (x) = x^{3} - 7 x + 7$ .
(a)
Show that $f (x) = 0$ has a root in $[- 3.1, - 3]$ .
[2 marks]
(b)
Find $f (1)$ and $f (2)$ . Explain why these alone do not show whether there are roots between $1$ and $2$ .
[2 marks]
(c)
Find the minimum value of $f (x)$ for $x > 0$ and hence show that there are two roots in $[1, 2]$ .
[4 marks]
(d)
Show that the smaller of these two roots lies in $[1.35, 1.36]$ .
[2 marks]
Total for D4: 10 marks
D5
The temperature of a cup of tea, $θ$ in °C above the room, $t$ minutes after it is made, is modelled by $θ = 70 e^{- 0.1 t} - 10$ . A second model for a cup that is reheated gives $θ = 4 t$ . The two models give the same temperature at $t = T$ .
(a)
Show that $T$ lies in $[6.5, 6.6]$ .
[3 marks]
(b)
Show that $T = 6.57$ to 2 decimal places.
[3 marks]
(c)
Find the temperature above the room at this time.
[2 marks]
Answer
(d)
Show that the two models agree at only one time.
[2 marks]
Total for D5: 10 marks
D6
The graphs of $y = 1.5 e^{- 0.5 x}$ and $y = ∣ x - 2∣$ are shown. Let $f (x) = 1.5 e^{- 0.5 x} - ∣ x - 2∣$ .
(a)
Explain how the graph shows that $1.5 e^{- 0.5 x} = ∣ x - 2∣$ has three roots, and say roughly where each one is.
[2 marks]
(b)
Show that one root lies in $[2.4, 2.5]$ .
[3 marks]
(c)
Find the negative root to 2 decimal places, showing that your answer is correct.
[3 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$f (x) = x^{5} - 5 x + k$ , where $k$ is a constant. Find the set of values of $k$ for which $f (x) = 0$ has three real roots. For $k = 1$ , find an interval of width $0.1$ containing each root.
[9 marks]
Hint 1 · What to try
$f^{'} (x) = 5 x^{4} - 5$ , so the stationary points are at $x = \pm 1$ .
Hint 2 · The first line
Three roots need the maximum above the axis and the minimum below it: $f (- 1) > 0$ and $f (1) < 0$ .
Hint 3 · The full method
For $k = 1$ , one root is in $[- 2, - 1]$ , one in $[0, 1]$ and one in $[1, 2]$ . Narrow each with more sign changes.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can show that a root lies in an interval, stating the change of sign and that the function is continuous.
I can show that a root is correct to a given number of decimal places by testing the bounds.
I can explain why an asymptote gives a sign change with no root.
I can explain why an interval with no sign change may still contain roots.

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