Quiz Chapter 03.04: Newton-Raphson Method of Solving 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

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?