Real Analysis

Taking Derivatives

Find the derivative of the SRK equation of state with respect to the compressibility factor \(Z\).

\[f (Z) = Z^3 - Z^2 + (A - B - B^2) Z - AB\] \[f' (Z) = 3Z^3 - 2Z + (A - B - B^2)\]

Error Bounds

For our 5th-order Taylor approximation of \(\sin x \approx x - \frac{x^3}{3!} + \frac{x^5}{5!}\), find a bound \(x_b\) such that truncation error \(\mid E_5 (x_b) \mid \leq 0.01\).

\[|E_n (x)| \leq \frac{M {|x - a|} ^ {n + 1}}{(n + 1)!}\] \[0.01 \leq \frac{|x_b - 0|^6}{6!}\] \[0.01 (6!) = {(x_b)}^6\] \[|x_b| \approx 1.39\] 
from math import factorial
import numpy as np

x = 1.39 # 5th-order bound xb
sinish_x = x - x**3 / factorial(3) + x**5 / factorial(5)
sin_x = np.sin(x)
print(f'True error = {sinish_x - sin_x:.4f}')

True error = 0.0019

Stability Bounds

sinish_x = x - x**3 / factorial(3) + x**5 / factorial(5)

Find bounds \(x_b\) such that, within these bounds, a given change in $x$ always produces an equal or smaller change in sinish(x).

\[\frac{\partial {\textbf{sinish} x}}{\partial x} \geq 1 - \frac{3x^2}{3!} + \frac{5x^4}{5!}\] \[1 \geq 1 - \frac{x_b^2}{2!} + \frac{x_b^4}{4!}\] \[\frac{x_b^4}{4!} \leq \frac{x_b^2}{2!}\] \[x_b^2 \leq 12\] \[x_b \leq \sqrt{12}\] \[x_b \leq 2 \sqrt 3\]

Let \(f (x) = 1.013 x^5 - 5.262 x^3 - 0.01732 x^2 + 0.8389 x - 1.912\).

  1. Evaluate \(f (2.279)\) by first calculating \({(2.279)}^2\), \({(2.279)}^3\), \({(2.279)}^4\), and \({(2.279)}^5\) using four-digit rounding arithmetic.

  2. Evaluate \(f (2.279)\) using the formula \(f (x) = (((1.013 x^2 - 5.262) x - 0.01732) x + 0.8389) x - 1.912\) and four-digit rounding arithmetic.

  3. Compute the absolute and relative errors in (1) and (2).