A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 10: Numerical methods · Lesson 3

The Newton–Raphson method

112 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Find where the tangent to $y = x^{3} - 5$ at the point where $x = 2$ crosses the $x$ -axis.
[3 marks]
Answer
A2
Needed today
Given $f (x) = x e^{x}$ , find $f^{'} (x)$ and the exact value of $f^{'} (1)$ .
[2 marks]
Answer
A3
Common slip
Given $f (x) = 3 cos x - x$ , find $f^{'} (1)$ to 4 decimal places. Use radians.
[2 marks]
Answer
A4
Needed today
Show that $x^{3} - 2 x - 5 = 0$ has a root between $2$ and $2.1$ .
[3 marks]

Example, then your turn

A-Level

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

Example 1

The diagram shows $y = f (x)$ , where $f (x) = x^{3} - 2 x - 5$ , and the tangent at $x_{0} = 3$ . The tangent meets the $x$ -axis at $x_{1}$ . Show that $x_{1} = x_{0} - \frac{f ( x _{0} )}{f ^{'} ( x _{0} )} = 2.36$ . Then use $x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}$ to find $x_{2}$ , $x_{3}$ and $x_{4}$ to 4 decimal places.

Hint The tangent has gradient

f^{'} (3) = 25

and passes through

(3, 16)

. It drops

16

over a horizontal distance of

\frac{16}{25}

. Then put the formula into the calculator with Ans and press = again and again.

B1
$f (x) = x^{3} + 3 x - 7$ . Use the Newton–Raphson method with $x_{0} = 1.5$ to find $x_{1}$ and $x_{2}$ to 4 decimal places.
[4 marks]
Answer

Example 2

$f (x) = x - 2 sin x$ . Use the Newton–Raphson method with $x_{0} = 2$ to find a root of $f (x) = 0$ to 4 decimal places.

Hint Set the calculator to radians before you start.

f^{'} (x) = 1 - 2 cos x

. Keep going until two iterates agree to 4 decimal places.

B2
$f (x) = 3 cos x - x$ . Use the Newton–Raphson method with $x_{0} = 1$ to find $x_{1}$ and $x_{2}$ to 4 decimal places. Use radians.
[4 marks]
Answer

Example 3

$f (x) = x e^{x} - 3$ . Use the Newton–Raphson method with $x_{0} = 1$ to find $x_{1}$ and $x_{2}$ to 4 decimal places.

Hint Differentiate

x e^{x}

with the product rule first:

f^{'} (x) = (x + 1) e^{x}

B3
$f (x) = x^{2} ln x - 2$ , $x > 0$ . Find $f^{'} (x)$ . Use the Newton–Raphson method with $x_{0} = 1.5$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[5 marks]
Answer

Example 4

The equation $e^{x} - 4 x = 0$ has a root $α$ near $2$ . Use the Newton–Raphson method with $x_{0} = 2$ to find $x_{1}$ , $x_{2}$ , $x_{3}$ . Show that $α = 2.153$ to 3 decimal places.

Hint The iterates suggest the value. The proof is a change of sign at the bounds

2.1525

and

2.1535

B4
The equation $ln x + x^{2} - 5 = 0$ has a root $β$ . Use the Newton–Raphson method with $x_{0} = 2$ to find $x_{1}$ and $x_{2}$ to 4 decimal places. Show that $β = 2.067$ to 3 decimal places.
[5 marks]
Answer

Example 5

$f (x) = x^{2} - 4 x + 1$ . Explain why $x_{0} = 2$ cannot be used to find a root with the Newton–Raphson method. Use $x_{0} = 3$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ .

Hint Work out

f^{'} (2)

. What does the tangent at a stationary point look like?

B5
$f (x) = x^{3} - 12 x + 5$ . Explain why the Newton–Raphson method fails with $x_{0} = 2$ . Use $x_{0} = 3$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[4 marks]
Answer

Practice

A-Level

C1
Sam uses the Newton–Raphson method on $f (x) = sin x + x - 2$ with $x_{0} = 1$ . His calculator is in degrees, and he gets $x_{1} = 1.4913$ and $x_{2} = 1.7327$ . Explain what has gone wrong, then find the correct $x_{1}$ and $x_{2}$ to 4 decimal places.
[5 marks]

C2
Jo wants the root of $x^{3} + x - 5 = 0$ near $1.5$ . She uses $x_{n + 1} = x_{n} + \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}$ with $x_{0} = 1.5$ and finds $x_{1} = 1.4839$ , $x_{2} = 1.4512$ , $x_{3} = 1.3838$ . Explain her mistake and find the correct $x_{1}$ and $x_{2}$ to 4 decimal places.
[4 marks]
Answer
C3
Kai uses the Newton–Raphson method on $f (x) = x e^{2 x} - 4$ with $x_{0} = 0.7$ . He writes $f^{'} (x) = 2 e^{2 x}$ and gets $x_{1} = 0.8432$ . Find his mistake. Find the correct $f^{'} (x)$ and the correct $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[5 marks]
Answer
C4
The equation $x^{3} - 3 x - 1 = 0$ has a root near $1.88$ . Lena uses the Newton–Raphson method with $x_{0} = 0.9$ to find it. Find $x_{1}$ to 4 decimal places. Explain, with reference to $f^{'} (x)$ , why her first step goes so far, and state which root her iterates then approach.
[5 marks]

Exam-style questions

Edexcel exam style

D1
$f (x) = e^{0.5 x} - x^{2} + 2$ . The equation $f (x) = 0$ has a root $α$ between $2$ and $3$ .
(a)
Show that $α$ lies between $2.2$ and $2.3$ .
[2 marks]
(b)
Find $f^{'} (x)$ .
[2 marks]
Answer
(c)
Using $x_{0} = 2.2$ , apply the Newton–Raphson method twice to find $x_{1}$ and $x_{2}$ to 4 decimal places.
[3 marks]
Answer
(d)
Show that $α = 2.256$ to 3 decimal places.
[3 marks]
Total for D1: 10 marks

D2
Ola writes: "The equation $x^{4} - 4 x - 1 = 0$ has a root between $1$ and $2$ . I will start the Newton–Raphson method at $x_{0} = 1$ , the lower end."
Ola's plan does not work.
(a)
Explain why the Newton–Raphson method cannot start at $x_{0} = 1$ .
[2 marks]
(b)
Use $x_{0} = 1.5$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[3 marks]
Answer
(c)
Show that the root is $1.663$ to 3 decimal places.
[2 marks]
Total for D2: 7 marks
D3
The equation $e^{- x} + x^{2} = 3$ has a positive root $α$ . Starting with $x_{0} = 1.5$ , use the Newton–Raphson method to find $α$ correct to 3 decimal places. Show that your answer is correct to this accuracy.
[7 marks]
Answer
D4
A drone flies above a lake. Its height, $h$ metres, $t$ seconds after it leaves a jetty is modelled by $h = 25 sin (\frac{t}{8}) + 2 - 0.8 t$ , where the angle is in radians. The drone reaches the water at time $T$ seconds.
(a)
Show that $T$ lies between $20$ and $21$ .
[2 marks]
(b)
Find $\frac{d h}{d t}$ .
[2 marks]
Answer
(c)
Use the Newton–Raphson method with $t_{0} = 20$ to find $T$ to 2 decimal places. Show that your answer is correct.
[3 marks]
Answer
(d)
Use the model to find the greatest height of the drone, to 3 significant figures.
[4 marks]
Answer
Total for D4: 11 marks
D5
The diagram shows $y = f (x)$ , where $f (x) = x^{3} - 3 x^{2} + 1$ . The curve has turning points at $(0, 1)$ and $(2, - 3)$ , and crosses the $x$ -axis three times.
(a)
Explain why $x_{0} = 2$ cannot be used to find the largest root with the Newton–Raphson method.
[2 marks]
(b)
Find $x_{1}$ when $x_{0} = 2.1$ , to 4 decimal places. Explain why $x_{1}$ is so far from $x_{0}$ .
[3 marks]
(c)
Use $x_{0} = 3$ to find the largest root to 4 decimal places.
[2 marks]
Answer
(d)
Starting from $x_{0} = 1$ , find $x_{1}$ and $x_{2}$ and state which root the iterates approach.
[2 marks]
Answer
Total for D5: 9 marks
D6
$f (x) = x^{3} - 4 x - 2$ . The equation $f (x) = 0$ has a root $α$ near $2$ .
(a)
Show that the Newton–Raphson formula for this equation can be written $x_{n + 1} = \frac{2 x _{n}^{3} + 2}{3 x _{n}^{2} - 4}$ .
[3 marks]
(b)
Use this formula with $x_{0} = 2$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[3 marks]
Answer
(c)
Show that $α = 2.214$ to 3 decimal places.
[2 marks]
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
The functions $f (x) = 3 x - 2$ and $g (x) = (x - 2)^{3}$ both have the root $x = 2$ . Apply the Newton–Raphson method to each. For $f$ , start at $x_{0} = 2.1$ , find $x_{1}$ , $x_{2}$ , $x_{3}$ , and show that the iterates move away from $2$ for every starting value other than $2$ . For $g$ , start at $x_{0} = 3$ and find the smallest $n$ for which $x_{n}$ is within $0.001$ of $2$ .
[9 marks]
Hint 1 · What to try
Write $e_{n} = x_{n} - 2$ , the distance from the root. Simplify $\frac{f ( x )}{f ^{'} ( x )}$ first.
Hint 2 · The first line
For $f$ , $\frac{f ( x )}{f ^{'} ( x )} = 3 (x - 2)$ . For $g$ , $\frac{g ( x )}{g ^{'} ( x )} = \frac{x - 2}{3}$ .
Hint 3 · The full method
Find how $e_{n + 1}$ is related to $e_{n}$ in each case. For $g$ you need $(\frac{2}{3})^{n} < 0.001$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the Newton–Raphson formula to find a root, using the full calculator value each time.
I can differentiate the function correctly first, with radians for trigonometric functions.
I can show that a root is correct to a stated accuracy with a change of sign.
I can explain why the method fails at or near a stationary point.

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