Lab 09 (Part - II): Problem Set

Solve the problems below using the library/package you created earlier in this lab.

  1. Solve the following equation $3x^3 - 4x^2 - 7x + 3.97 = 0$ using bisection and secant methods with appropriate initial values. How many iterations did each method take?
  2. Solve the following equation using Newton method: $3.82x^2 - 11.9x + 2.223 = 0$. Change the initial value three times and for each value, note down how many iterations are required for Newton method.
  3. Find the area under the curve $y = 9.3x^4 - 1.65x^3 + 14.78x - 2.43$ and the line $y = 12.6x - 4$ between the points $x = 1$ to $x = 2.5$ Use Trapezoidal rule with $h = 0.25$.
  4. Find the length of the curve $3.2x^2 - 7.2x + 17.12$.
  5. Experimental data is given in the Table below. Do you think that there is a simple linear relationship between the $X$ and $Y$ values? If so, find the relationship between them. $$ \begin{array}{|c|c|} \hline\hline X & Y \\ \hline\hline -1 & 36.7 \\ -0.5 & 38.2 \\ 0 & 39.1 \\ 1 & 34.8 \\ 2 & 33.5 \\ 2.5 & 29.6 \\ 3 & 29.25 \\ 4 & 26.75 \\ \hline \end{array} $$
  6. Solve the system of equations given by $$ \begin{eqnarray*} -0.793x_1 + 2.396x_2 + 2.272x_3 + 3.763x_4 & = & 47.56 \\ 2.609x_1 + 0.119x_2 + 0.323x_3 + 0.193x_4 & = & 7.918 \\ 1.773x_1 - 0.651x_2 + 2.967x_3 + 2.095x_4 & = & 18.72 \\ -1.39x_1 + 3.669x_2 + 2.481x_3 + 2.824x_4 & = & 47.975 \end{eqnarray*} $$
  7. Temperature is measured at different distances from a heat source and the data is given in the table below. Distances are in metres and temperatures are in Celsius. Find the temperature at a distance of $3.75\,$m from the source. $$ \begin{array}{||l||c|c|c|c|c|c|c|c|c|} \hline D & 1. & 1.5 & 2. & 2.5 & 3. & 3.5 & 4. & 4.5 & 5.\\ \hline T & 100.63& 61.03& 37.02 & 22.45& 13.62& 8.26& 5.01& 3.04& 1.84 \\ \hline \end{array} $$