Use the bisection method to find the root of the function $f(x) = e^{-x} - x$. Perform three iterations using an initial bracket of $x_l = 0$ and $x_u = 1$. Calculate the approximate relative error for each iteration.

The velocity of a falling parachutist is given by $v(t) = \frac{gm}{c} \left(1 - e^{-(c/m)t}\right)$. Given $v = 40 \text{ m/s}$, $m = 68.1 \text{ kg}$, $t = 10 \text{ s}$, and $g = 9.8 \text{ m/s}^2$, use the bisection method to determine the drag coefficient $c$ to a level of $\epsilon_a \leq 5\%$. The initial guesses are $c_l = 12$ and $c_u = 16$.

Use the false position method to determine the root of $f(x) = -0.5x^2 + 2.5x + 4.5$. The initial interval is $[5, 10]$. Perform two iterations.

Use the Newton-Raphson method to estimate the root of $f(x) = e^{-x} - x$. Employ an initial guess of $x_0 = 0$. Perform three iterations.

Use the Secant method to estimate the root of $f(x) = e^{-x} - x$. Start with initial estimates of $x_{-1} = 0$ and $x_0 = 1$. Perform two iterations.

Find the root of $f(x) = (x - 3)(x - 1)(x - 1) = x^3 - 5x^2 + 7x - 3$, which has a double root at $x=1$. Use the Modified Newton-Raphson method with an initial guess $x_0 = 0$. Perform one iteration.

Solve the following system using the multivariable Newton-Raphson method with an initial guess of $x_0 = 1.5, y_0 = 2.0$. Perform one iteration.

$f_1(x, y) = x^2 + xy - 10 = 0$

$f_2(x, y) = y + 3xy^2 - 57 = 0$

Find the roots of the quadratic polynomial extracted from Bairstow's method: $x^2 - 2x + 5 = 0$.