Locating roots and iterationEdexcel A-Level Maths: Revision notes

If $f$ is continuous on $[a,b]$ and $f(a)$, $f(b)$ have opposite signs, then $f(x)=0$ has at least one root in $[a,b]$. Example: $f(x)=x^3-3x-5$ has $f(2)=-3$ and $f(3)=13$, so a root lies in $[2,3]$. Narrowing further, $f(2.2)=-0.952$ and $f(2.3)=0.267$ show $2.2<\alpha<2.3$. Always state: the function is continuous, the values (or signs) at both ends, and the conclusion.

The method can fail in two ways.

• An interval that is too large may contain an even number of roots, so the signs at the ends match. For $g(x)=(x-2)(x-3)$, $g(1.5)=g(3.5)=0.75$, yet the interval contains two roots. A narrower interval such as $[1.5,2.5]$ shows the sign change: $g(2.5)=-0.25$.
• A discontinuous function can change sign across an asymptote without a root. $f(x)=\frac{1}{x-2}$ has $f(1)=-1$, $f(3)=1$, but is never zero. A root where the curve touches the axis without crossing also gives no sign change.

To solve $f(x)=0$ approximately, rearrange it as $x=g(x)$ and iterate $x_{n+1}=g(x_n)$ from a starting value $x_0$. For $x^3-3x-5=0$, use $x=\sqrt[3]{3x+5}$: $x_0=2$, $x_1=2.2240$, $x_2=2.2684$, $x_3=2.2770$, $x_4=2.2786$. The values settle at $\alpha=2.28$ (2 d.p.). For $\mathrm{e}^x=4x$, two rearrangements $x=\ln(4x)$ and $x=\frac{\mathrm{e}^x}{4}$ find different roots. Use the answer button on your calculator ($\text{Ans}$) to repeat the step, and keep full calculator accuracy.

Draw $y=g(x)$ and $y=x$. Start at $x_0$ on the $x$-axis, go vertically to the curve, then horizontally to the line $y=x$, and repeat.

• If $0<g'(x)<1$ near the root, the path is a staircase moving monotonically towards the root.
• If $-1<g'(x)<0$, the path spirals in as a cobweb.
• If $|g'(x)|>1$, the path moves away: the iteration diverges. Example: for $g(x)=\frac{\mathrm{e}^x}{4}$, $g'(x)=\frac{\mathrm{e}^x}{4}$, which equals $x$ at a root, so it converges to $\alpha\approx0.36$ ($g'<1$) and cannot find $\beta\approx2.15$ ($g'>1$).

Many equations, such as $\mathrm{e}^x=4x$ or $x^3-3x-5=0$, cannot be solved by algebra. Numerical methods give a root to any required accuracy. To show a root is correct to $n$ decimal places, find a sign change over an interval of $\pm5$ in the next decimal place: $f(2.145)<0$ and $f(2.155)>0$ show $\beta=2.15$ to 2 d.p. State the accuracy asked for and show the values you used.