# Quiz Chapter 03.04: Newton-Raphson Method of Solving Nonlinear Equations

 MULTIPLE CHOICE TEST NEWTON-RAPHSON METHOD NONLINEAR EQUATIONS

1. The Newton-Raphson method of finding roots of nonlinear equations falls under the category of _____ methods.

2. The Newton-Raphson method formula for finding the square root of a real number $R$ from the equation $x^{2}-R=0$ is

3. The next iterative value of the root of $x^{2} - 4 = 0$ using Newton-Raphson method, if the initial guess is $3$, is

4. The root of the equation $f(x) = 0$ is found by using the Newton-Raphson method. The initial estimate of the root is $x_{0} = 3$, and $f \left( 3 \right) = 5$. The angle the line tangent to the function $f(x)$ makes at $x=3$ with the $x$-axis is $57^{\circ}$. The next estimate of the root, $x_{1}$ most nearly is

5. The root of $x^{3} = 4$ is found by using the Newton-Raphson method. The successive iterative values of the root are given in the table below

 Iteration Number Value of Root $0$ $2.0000$ $1$ $1.6667$ $2$ $1.5911$ $3$ $1.5874$ $4$ $1.5874$

The iteration number at which I would find trust in at least two significant digits in the answer is

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 $v$?