Assignment - 2 (Due: 13 November 2017)

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1. Solve the following system of equations: $$\begin{eqnarray*} 3x^2 + 4x - y & = & 7 \\ 2x - y & = & -1 \end{eqnarray*}$$
2. Let $$f(x) = \int_{-1}^{2}\frac{x}{1+x^4}\,dx$$
   • Use midpoint rule, trapezoidal rule and Simpson's rule to compute the integral. Are all the values the same? If the exact answer is $0.27020975\ldots$, which method gave the most accurate result?
   • What is the result accurate to 10 decimal places? How small a $h$ does it require?
   • Draw a plot of accuracy (in decimal places) vs. $h$ for $1, 2, \ldots, 12$ decimal places. Analyse and comment on your findings for Simpson's 1/6 and 3/8 rules.
3. Find $\sqrt[5]{100}$ accurate to 10 decimal places.
4. Find the solution of $$\frac{d^2y}{dx^2} - 4x = 1.5$$ with boundary values of $y(0)=y(2.5)=0$. What happens as you change the second boundary condition from $y(2.5)=0$ to $y(2.5)=50$? What happens if you change the first condition similarly?
5. Find at least two roots of the function $$y = 4.3\frac{\sin(2x)}{2x}$$ in the interval $(3,8)$ using bisection and secant methods with different initial guesses. It will be clear that the secant method does not behave as `nicely' as bisection method for some guesses. Clearly explain the behaviour of secant method with the help of diagrams, as necessary.