Quiz Chapter 03.03: Bisection Method of Solving a Nonlinear Equation

1. The bisection method of finding roots of nonlinear equations falls under the category of a(n) _____ method.

2. If for a real continuous function $f(x)$, you have $f(a) f(b) < 0$, then in the interval $\left[ a,b \right]$ for $f(x) = 0$, there is (are)

3. Assuming an initial bracket of $\left[ 1,5 \right]$, the second (at the end of $2$ iterations) iterative value of the root $te^{-t} - 0.3 = 0$ is

4. To find the root of $f(x)= 0$, a scientist uses the bisection method. At the beginning of an iteration, the lower and upper guesses of the root are $x_{l}$ and $x_{u}$ respectively. At the end of this iteration the absolute relative approximate error in the estimated value of the root would be

5. For an equation like $x^{2}=0$, a root exists at $x=0$. The bisection method cannot be adopted to solve this equation in spite of the root existing at $x=0$ because the function $f(x)= x^{2}$

6. The Ideal Gas Law is given by

$pv=RT$

where $p$ is the pressure, $v$ is the specific volume, $R$ is the universal gas constant, and $T$ is the absolute temperature. This equation is only accurate for a limited range of pressure and temperature. Vander-Waals came up with an equation that was accurate for larger range of pressure and temperature given by

$\left( p + \dfrac{a}{v^{2}} \right) \left( v-b \right) = RT$

where $a$ and $b$ are empirical constants dependent on a particular gas. Given the value $R=0.08$, $a=3.592$, $b=0.04267$, $p=10$ and $T=300$ (assume all units are consistent), one is going to find the specific volume, $v$, for the above values. Without finding the solution from the Vander-Waals equation, what would be a good initial guess for $v$?