Sketch graphs to show root existence

A question is this type if and only if it asks to sketch suitable graphs to demonstrate that an equation has exactly one (or a specific number of) real root(s).

29 questions · Standard +0.1

1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams
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CAIE P3 2018 November Q3
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation \(x^3 = 3 - x\) has exactly one real root. [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$ converges, then it converges to the root of the equation in part (i). [2]
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
OCR C3 Q3
12 marks Moderate -0.3
  1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x^5 \quad \text{and} \quad y = a - bx$$ on the same diagram, show that the equation $$x^5 + bx - a = 0$$ has exactly one real root. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[5]{53 - 2x_n}\), with a suitable starting value, to find the real root of the equation \(x^5 + 2x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places. [4]
OCR C3 2013 January Q6
11 marks Standard +0.3
  1. By sketching the curves \(y = \ln x\) and \(y = 8 - 2x^2\) on a single diagram, show that the equation $$\ln x = 8 - 2x^2$$ has exactly one real root. [3]
  2. Explain how your diagram shows that the root is between 1 and 2. [1]
  3. Use the iterative formula $$x_{n+1} = \sqrt{4 - \frac{1}{2}\ln x_n},$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places. [4]
  4. The curves \(y = \ln x\) and \(y = 8 - 2x^2\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places. [3]
AQA Paper 1 2019 June Q7
11 marks Standard +0.3
  1. By sketching the graphs of \(y = \frac{1}{x}\) and \(y = \sec 2x\) on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for \(x > 0\) [3 marks] \includegraphics{figure_7a}
  2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
  3. Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
    1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with \(x_1 = 0.4\), to find \(x_2\), \(x_3\) and \(x_4\), giving your answers to four decimal places. [2 marks]
    2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\), \(x_3\) and \(x_4\). [2 marks] \includegraphics{figure_7d}