Iterative formula with graphical justification

Questions asking to show graphically that an equation has a certain number of roots, then use an iterative formula to find a root to specified accuracy.

4 questions

CAIE P2 2022 June Q5
5
  1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
  2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
  3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
  4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2002 November Q4
4
  1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\sin x = \frac { 1 } { x ^ { 2 } }$$
  2. Verify by calculation that this root lies between 1 and 1.5.
  3. Show that this value of \(x\) is also a root of the equation $$x = \sqrt { } ( \operatorname { cosec } x )$$
  4. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$ to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.
OCR C3 2011 June Q4
4
  1. Show by means of suitable sketch graphs that the equation $$( x - 2 ) ^ { 4 } = x + 16$$ has exactly 2 real roots.
  2. State the value of the smaller root.
  3. Use the iterative formula $$x _ { n + 1 } = 2 + \sqrt [ 4 ] { x _ { n } + 16 }$$ with a suitable starting value, to find the larger root correct to 3 decimal places.
OCR C3 2012 June Q5
5
  1. It is given that \(k\) is a positive constant. By sketching the graphs of $$y = 14 - x ^ { 2 } \text { and } y = k \ln x$$ on a single diagram, show that the equation $$14 - x ^ { 2 } = k \ln x$$ has exactly one real root.
  2. The real root of the equation \(14 - x ^ { 2 } = 3 \ln x\) is denoted by \(\alpha\).
    (a) Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
    (b) Use the iterative formula \(x _ { n + 1 } = \sqrt { 14 - 3 \ln x _ { n } }\), with a suitable starting value, to find \(\alpha\). Show the result of each iteration, and give \(\alpha\) correct to 2 decimal places.