9.5. Equations

Straight line

\[ y = mx + b \]

\[ Ax + By + C = 0 \]

Point-slope formula

\[ y - y_1 = m(x-x_1) \]

\[ m = (y_2 - y_1)/(x_2 - x_1) \]

Angle between lines

\[ \alpha = arctan[(m_2 - m_1)/(1 + m_2 m_1)] \]

Distance between two points

\[ d = \sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2} \]

Quadratic Equation

\[ x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a} \]

Mean of a population

\[ \mu = \frac{1}{N}\sum_{i=1}^N \]

Mean of a sample

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^n \]

Standard deviation of a sample

\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2} \]

Standard deviation of a population

\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2} \]

Variance of a sample

\[ s^{2} = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2 \]

Variance of a population

\[ \sigma^{2} = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2 \]

Z-transform

\[ z = \frac{x-\mu}{\sigma} \]

Ohm’s Law

\[ V = IR \]

\[ I = \frac{V}{R} \]

\[ R = \frac{V}{I} \]

Resistors in series

\[ R_t = R_1 + R_2 + R_3 + \ldots + R_n \]

\[ I_t = I_1 = I_2 = I_3 = \ldots = I_n \]

\[ V_t = V_1 + V_2 + V_3 + \ldots + V_n \]

Resistors in parallel

\[ \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n} \]

\[ V_t = V_1 = V_2 = V_3 = \ldots = V_n \]

\[ I_t = I_1 + I_2 + I_3 + \ldots + I_n \]

Power in steady-state DC circuits

\[ P = VI \]

\[ P = I^{2}R \]

\[ P = \frac{V^2}{R} \]