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1. Show that $$\sqrt[3]{x}-\cos x - 1 = 0$$ has a root in the interval $[1.4,1.5]$ and find the root correct to 3 decimal places.
2. The function $$\mathrm{f}(x) = x(x+2)(x-2)$$ has three roots. Show on a suitable diagram which of the three roots will be obtained when using the Newton-Raphson method with $x_0 = 1.5$, and state two values of $x_0$ which will cause the Newton-Raphson procedure to fail.
3. Show that $$\mathrm{f}(x) = 2 + \tan x$$ changes sign in the interval $[1.5,1.6]$, and state, with a reason, whether $\mathrm{f}(x)$ has a root in this interval.
4. A student observes that there is a change of sign on the interval $[-0.5,1]$ for $$\mathrm{f}(x) = \dfrac{1}{x^3+x^2} - 1$$ The student writes:
There are no roots in this interval because there is a discontinuity in the interval.
Has the student made the correct conclusion?
5. Given $\mathrm{f}(x) = x^2-x-1$,
  1. Show that $\mathrm{f}(x) = 0$ can be rewritten in the form $x = \sqrt{x+1}$.
  2. Use the iterative formula $x_{n+1} = \sqrt{x_n+1}$ with $x_0 = 0.5$ to find an approximation to the solution to $\mathrm{f}(x)=0$ to 2 decimal places, and draw a staircase diagram to illustrate this process.
6. Given $\mathrm{f}(x) = x^2-x-1$,
  1. Show that $\mathrm{f}(x) = 0$ can be rewritten in the form $x = \dfrac{1}{x-1}$.
  2. Use the iterative formula $x_{n+1} = \dfrac{1}{x_n-1}$ with $x_0=-2$ to find $x_1$, $x_2$, and $x_3$, and draw a cobweb diagram to illustrate this process.
7. Given $$\mathrm{f}(x) = x^2-x-1$$ show, on a suitable diagram, that the iterative formula $x_{n+1} = x^2_n - 1$ does not give a solution to the equation $\mathrm{f}(x) = 0$ when $x_0 = 2$, and explain why this is the case.
8. Use $x_0 = 1.2$ and the Newnton-Raphson method to find a second approximation to the root of the equation $$1 - e^x + 3\sin 2x = 0$$ and show that this answer is correct to 2 decimal places.
9. Show that there is a root of the equation $$\cos x - x^2 = 0$$ in the interval $[0.8,0.9]$, and find the root to 3 decimal places using an appropriate iterative formula.
10. Use the Newton-Raphson method twice, starting with $x_1 = 0.5$, to find, correct to 4 decimal places, an approximation for a root of the equation $$x^3 + 10x - 4 = 0$$
11. For $\sqrt{x}-\frac{2}{x} = 0$,
  1. Show that the solution is $1.587$ correct to 3 decimal places.
  2. Write down the exact value of the solution.
12. Given $\mathrm{f}(x) = x^4 - 21x - 18$,
  1. Show that $-0.834$ is a solution to the equation $\mathrm{f}(x) = 0$ correct to 3 decimal places.
  2. Find the $x$-coordinates of any stationary points of $y=\mathrm{f}(x)$, correct to 2 decimal places.
13. Given $\mathrm{f}(x) = e^{x-1} + 2x - 15$,
  1. Show that the equation $\mathrm{f}(x) = 0$ can be written as $x = \ln(15-2x) + 1$.
  2. Use this as the basis of an iterative formula, with $x_0 = 3$, to find $x_1$ and $x_2$ to 4 decimal places.
  3. Choose a suitable interval to show that the root of the equation $\mathrm{f}(x) = 0$ is $3.16$ to 2 decimal places.
14. Given $\mathrm{f}(x) = x^2-6x+1$,
  1. Show that the equation $\mathrm{f}(x) = 0$ can be written as $x=\dfrac{x^2+1}{6}$
  2. Use a diagram to explain why $x_0 = 2$ gives a root of the equation $\mathrm{f}(x) = 0$ using the iterative formula $x_{n+1} = \dfrac{x^2_n+1}{6}$
  3. $\mathrm{f}(x) = 0$ has a second solution at $x \approx 6$. Explain why this iterative formula can never find the second solution.
15. Given $\mathrm{f}(x) = 4\cot x - 8x + 3$,
  1. Show that there is a root, $\alpha$, of $\mathrm{f}(x) = 0$ in the interval $[0.8,0.9]$.
  2. Show that the equation $\mathrm{f}(x) = 0$ can be written in the form $x = \dfrac{\cos x}{2\sin x} + \dfrac{3}{8}$.
  3. Use this as the basis of an iterative formula with $x_0 = 0.85$ to calculate $x_1$ and $x_2$
  4. By considering the change of sign of $\mathrm{f}(x)$ in a suitable interval, show that $\alpha =0.831$ to 3 decimal places.
16. For the equation $\sin x - \cos 2x - 0.4 = 0$,
  1. Show that there is a root in the interval $[0.5,1]$.
  2. Use the iterative formula $x_{n+1}=\dfrac{\mathrm{arccos}(\sin x_n - 0.4)}{2}$ with $x_0 = 0.6$, writing down all the digits on your calculator, to find $x_1$, $x_2$ and $x_3$.
17. Given $\mathrm{f}(x) = x^3-2x^2-5x-4$,
  1. The graph of $y=\mathrm{f}(x)$ has a stationary point with $x$ coordinate $p$. Explain why $x_0=p$ is not suitable to use when applying the Newton-Raphson method.
  2. Use $x_0=3.5$ and apply the Newton-Raphson procedure twice to find an approximation to a root of the equation $\mathrm{f}(x)=0$.
  3. Show that this answer is accurate to 3 decimal places.
18. Given $\mathrm{f}(x) = x^2-\dfrac{3}{x^2}$,
  1. Show that the equation $\mathrm{f}(x)=0$ has a root $\alpha$ in the interval $[1.3,1.4]$
  2. Use 1.3 as a first approximation to $\alpha$ and apply the Newton-Raphson procedure once to find a second approximation to $\alpha$, giving your answer to 3 decimal places.
  3. Determine if this answer is correct to 3 decimal places.
19. The equation $2x^3+x^2-1=0$ has exactly one real root.
  1. Show that the Newton Raphson formula can be written $$x_{n+1} = \dfrac{4x_n^3+x_n^2+1}{6x_n^2+2x_n}$$
  2. Use $x_1=1$ to find $x_2$ and $x_3$.
  3. Explain why $x_1=0$ cannot be used with this formula.
20. The equation $$\mathrm{arcsin} (2x) - 0.5x - 0.7 = 0$$ can be rearranged to give the iterative form $$x_{n+1} = a\sin(bx_n+c)$$
  1. Find the values of the constants $a$, $b$ and $c$.
  2. Use this iterative formula with $x_0 = 0.4$ to find the solution to the equation correct to 3 decimal places.
21. The curve with equation $y = 2^x$ intersects the straight line with equation $y = 3 - 2x$ at the point $P$, whose $x$ coordinate is $\alpha$. Starting with $x = 0.5$, use the Newton-Raphson method to find $\alpha$ correct to 3 decimal places
22. For the equation $x^3 - 5x^2 + 12x - 6 = 0$,
  1. Show that $\alpha = 0.6556$, to 4 decimal places, is a root.
  2. A student attempts to find $\alpha$ by using the iteration $x_{n+1} = \sqrt[3]{5x_n^3-12x_n+6}$. Explain why the student will fail to find $\alpha$.
  3. Use $x_{n+1} = \dfrac{6+5x_n^2-x_n^3}{12}$ to find $x_1$, $x_2$ and $x_3$, starting from $x_0 = 0.6$.
23. $$\mathrm{f}(x) = 4\csc x - 4x + 1$$
  1. Show that there is a root $\alpha$ of $\mathrm{f}(x) = 0$ in the interval $[1.2,1.3]$.
  2. Show that the equation $\mathrm{f}(x) = 0$ can be written as $x = \dfrac{1}{\sin x} + \dfrac{1}{4}$.
  3. Use this as the basis of an iterative formula, with $x_0 = 1.25$, to find the values of $x_1$, $x_2$ and $x_3$ to four decimal places.
  4. By considering the change of sign in a suitable interval, verify that $\alpha = 1.291$ is correct to three decimal places.
24. Use the Newton-Raphson method once, starting with $0.9$ as the first approximation, to find a second approximation for a root of the equation $\dfrac{4x-3}{x^2+1}-\mathrm{e}^{-x} = 0$.
25. A curve with equation $x^3 + y = xy$ meets the line with equation $y + 3x + 1 = 0$ at the point $P$.
  1. Use an iterative method based on the formula $x_{n+1} = \dfrac{1}{2}(Ax_n^3 + Bx_n^2 + C)$, where $A$, $B$, and $C$ are constants to be found, and $x_1 = -0.35$, to find the $x$ coordinate of $P$ to 3 decimal places.
  2. Use the Newton-Raphson method with the same starting value to find the $x$ coordinate of $P$ and explain the advantage of the Newton-Raphson method.
26. The curve $C$ has equation $$(x-4)^2 + (y-1)^2 = 9$$ A student wants to find where the curve crosses the $x$ axis by rearranging it into $$y = 1 + \sqrt{9-(x-4)^2}$$ The student finds that there are no roots and concludes that $C$ does not cross the $x$ axis. Is the student correct?
27. A curve has equation $$(x-3)^2 + (y-2)^2 = 4$$
  1. Show that the equation of the curve can be written as $y = 2 \pm \sqrt{6x-x^2-5}$
  2. A student uses the sign change method to estimate the solution to the equation $2 - \sqrt{6x-x^2-5} = 0$, and determines that there is no solution in the interval $[2.9,3.1]$. Explain whether or not the student is correct.
28. A student claims there are no roots of the equation $$\sin(\sqrt{x}) + \sin(x^2) = 0$$ in the interval $[2.1,2.2]$ by using the sign change method. By finding the roots of this equation in this interval, explain why the sign change method failed in this instance.
29. The curve $x^3 + xy + y^3 = 10$ meets the line $y = x+2$ at the point $A$. Use the Newton-Raphson method, starting at $x = 0.1$, to find the $x$ coordinate correct to 3 decimal places.
30. The curve $$y = \sin(2x) + \ln x$$ has a stationary point around $x = 1.05$. Use the Newton-Raphson method to find an approximation for the $x$ coordinate of the stationary point correct to 3 decimal places.