Problem: The aim of this question is to assess your understanding of the contraction mapping theorem and various methods for finding the root of an equation.

Consider the equation

\[e^{x}-x^{3}=0 .\]

1. By sketching a graph, or otherwise, show that the equation has two roots in the interval \([-2,5]\).
2. Explain why the equation has no more roots.
3. Verify that there is one root in the interval [1.5, 2] and give its approximate value to one decimal place.
4. Prove that the iterative scheme
   \[x_{r+1}=g\left(x_{r}\right)=e^{x_{r} / 3}\]
   with a suitable starting point, converges to the root in [1.5,2], by showing that \(g\) is a contraction mapping on this interval.
5. Use your calculator and the iterative scheme in (d) to compute the root in [1.5, 2] to two decimal places, paying particular attention to the stopping criterion.
6. Explain why the iterative scheme in (d) would not be appropriate to use to try to find the larger root.
7. Use an appropriate Mathcad worksheet to find the larger root, accurate to six decimal places, using the Newton -Raphson and add- $n x$ methods. Compare the efficiencies of these models.

Problem: The aim of this question is to assess your ability to solve a non-linear equation and to assess your understanding of the bisection method.

A ball is fired in an indoor sports hall at \(30 \mathrm{~m} \mathrm{~s}^{-1}\) at an angle \(\alpha\) to the horizontal. It moves with air resistance proportional to its speed.

When it is at a horizontal distance \(x\) metres from the point of firing, its height \(y\) metres can be modelled by the equation

\[y=x \tan \alpha+\frac{x}{\cos \alpha}+90 \ln \left(1-\frac{x}{90 \cos \alpha}\right)\]

1. When \(\alpha=0.8\), use an appropriate Mathcad worksheet to show that there are two times when the ball is at a height of \(10 \mathrm{~m}\).
2. Find the first of these times, accurate to two decimal places, using the Newton-Raphson method.
3. A new ceiling was installed in the sports hall \(10 \mathrm{~m}\) above the ground.
   Show that if \(\alpha=0.4\) then the ball will not hit the ceiling.
4. Use an adaptation of the bisection method, with values of \(\alpha\) in the interval [0.4,0.8], to find, to two decimal places, the largest angle at which the ball can be fired so that it does not hit the ceiling.

Price: $19.83

Solution: The downloadable solution consists of 11 pages, 883 words and 9 charts.
Deliverable: Word Document