Slope of Tangent Lines, Special Limits, and Algebra Review

Vocabulary flashcards covering key algebraic rules, line formulas, tangent slopes, velocity definitions, and trigonometric limits from the lecture notes.

Negative Exponent Rule

The algebraic exponent rule stating that $a^{-n} = \frac{1}{a^n}$.

Zero Product Property

The algebraic property stating that if $A \times B = 0$, then either $A = 0$ or $B = 0$.

Quadratic Formula

The formula used to solve quadratic equations of the form $a x^2 + b x + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$.

Difference of Squares Formula

The algebraic factoring identity given by $a^2 - b^2 = (a - b)(a + b)$.

Square of a Binomial Formula

The expansion identity given by $(a + b)^2 = a^2 + 2 a b + b^2$.

Difference of Cubes Formula

The algebraic factoring identity given by $a^3 - b^3 = (a - b)(a^2 + a b + b^2)$.

Distance Formula

The formula used to calculate the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$, given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Slope-Intercept Form

The equation of a straight line given by $y = m x + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Point-Slope Form

The equation of a straight line given by $y - y_1 = m (x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

Secant Line

A straight line passing through two points on a curve, whose slope $m_{\text{sec}} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$ represents the average rate of change.

Tangent Line

A straight line touching a curve at a specific point, whose slope $m_{\text{tan}} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ represents the instantaneous rate of change at $x = a$.

Average Velocity

The ratio of change in position to change in time, calculated as $v_{\text{avg}} = \frac{h(t) - h(b)}{t - b}$, which corresponds to the slope of a secant line.

Instantaneous Velocity

The limit of average velocity as time $t$ approaches $b$, calculated as $v(b) = \lim_{t \to b} \frac{h(t) - h(b)}{t - b}$, which corresponds to the slope of a tangent line.

Special Limit of Sine over Angle

The fundamental trigonometric limit evaluated as $x$ approaches $0$, given by $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.

Special Limit of Cosine Difference over Angle

The trigonometric limit evaluated as $x$ approaches $0$, given by $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$.

Special Limit of Tangent over Angle

The trigonometric limit evaluated as $x$ approaches $0$, given by $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$.

Secant Trigonometric Identity

The reciprocal trigonometric identity defined as $\sec(x) = \frac{1}{\cos(x)}$.

Trigonometric Signs by Quadrant

The signs of sine, cosine, and tangent functions across Quadrants I, II, III, and IV.

Procedure for Finding Intercepts

To find the $x$-intercept of a function set $y = 0$, and to find the $y$-intercept set $x = 0$.