Approximating solutions to equations | Maths | Quizalize & Oak National Academy

Starter quiz

What is 0.0647399 rounded to 2 significant figures?

0.1

0.064

0.065 ✓

0.0647

6.47
What is the value of the expression $x^3 +3x-12$ when $x=2$ ?

'2' ✓
For which values of $x$ are we able to evalate the expression $\sqrt{6-2x}$ ?

-1 ✓

0 ✓

2 ✓

3 ✓

4
Use the iterative formula $P_{t+1}=2.5(P_t-50)$ with $P_0=200$ to find the value of $P_4$ . Write down your full calculator display.

'4640.625' ✓
Using the iterative formula $x_{n+1}=10-4(x_n)$ with $x_0=1$ which of these is the value of $x_8$ ?

$-22$

$16\;386$

$-65\;534$ ✓

$262\;146$
Using the iterative formula $x_{n+1}=\sqrt{10-2(x_n)}$ and $x_0=3$ match up the iterations with their values.

$x_1$

⇔

2 ✓

$x_2$

⇔

2.449489... ✓

$x_3$

⇔

2.258543... ✓

$x_4$

⇔

2.341561... ✓

$x_5$

⇔

2.305835 ... ✓

Exit quiz

Which value does the iterative formula $x_{n+1}=\frac{5}{x_n}-3$ converge on to 1 significant figure when $x_0=2$ ?

-10

-4 ✓

-3

-0.5
Which value does the iterative formula $x_{n+1}=\frac{5}{x_n}-3$ converge on to 2 significant figures when $x_0=2$ ?

-4.5

-4.4

-4.2 ✓

-4.1
How can we tell if an iterative formula is converging for a given value of $x_0$ ?

The iterations alternate between positive and negative.

The iterations are increasing each time.

The iterations are decreasing each time.

The difference between iterations is decreasing. ✓
Which of these could be an iterative formula to find a solution to $x^3-3x-5=0$ ?

$x_{n+1}=\frac{(x_n)^3}{3} - 5$

$x_{n+1}=\frac{(x_n)^3+5}{3}$

$x_{n+1}=\sqrt[3]{-3(x_n)-5}$

$x_{n+1}=\sqrt[3]{3(x_n)+5}$ ✓
To find a solution to the equation $x^3+6x-10=0$ the iterative formula $x_{n+1}=\frac{10-(x_n)^3}{6}$ can be used. Using $x_0=1$ find the value of $x_3$

-2.8333...

1.1041...

1.1666...

1.4423... ✓
Using the iterative formula $x_{n+1}=\frac{10-(x_n)^3}{6}$ and $x_0=1$ there is a solution to $x^3+6x-10=0$ correct to 2 significant figures when $x=$ ______

'1.3' ✓

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Lesson Details

Key learning points

You can rearrange an equation in order to carry out iteration
This process finds an approximate solution to the equation
This process is carried out until the outputs are the same to a given degree of accuracy
Efficient use of the calculator makes this process much faster

Common misconception

To find a solution to a given degree of accuracy an iteration can just be rounded.

Several successive iterations need to round to the same value to a given degree of accuracy. If in doubt efficient calculator use means many iterations can be generated easily until confident that a digit is fixed.

Keywords

Iteration - Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.