Applied Calculus Review Flashcards

Vocabulary practice flashcards covering non-linear inequalities, lines, piecewise functions, circles, ellipses, and economic functions.

Slope-Intercept Form

An equation for any non-vertical line written as $y = m x + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept.

Point-Slope Equation

An equation of a line with slope $m$ passing through a known point $(x_1, y_1)$, expressed as $y - y_1 = m(x - x_1)$.

Slope Formula

The formula used to calculate the slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$, given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.

General Linear Equation

An equation of the form $A x + B y = C$, where $A$, $B$, and $C$ are constants with $A$ and $B$ not both zero, whose graph is a straight line.

Perpendicular Lines

Two non-vertical lines that intersect at right angles ($90^\circ$), where the product of their slopes $m_1$ and $m_2$ is $-1$ ($m_1 m_2 = -1$).

Break-Even Point

The sales level or number of units at which total revenue equals total cost ($R(x) = C(x)$), resulting in a total profit of zero ($P(x) = 0$).

Straight-Line Depreciation

A method of writing off an asset's value where it depreciates by a fixed constant amount each year over its expected lifetime $T$.

Piecewise Linear Graph

A graph composed of multiple distinct linear segments defined over different sub-intervals of the domain.

Standard Form of a Circle

The equation $(x - h)^2 + (y - k)^2 = r^2$, representing a circle centered at the point $C(h, k)$ with radius $r$.

General Form of a Circle

An equation written as $x^2 + y^2 + a x + b y + c = 0$, which can represent a circle, a single point, or a contradiction after completing the square.

Tangent Line to a Circle

A straight line that intersects a circle at a single point $P$ on its circumference and is perpendicular to the radius drawn to $P$.

Tangent Line to a Smooth Curve

The line that provides the best linear approximation to a smooth curve near a given point of tangency $P$.

Ellipse

A closed curve centered at the origin defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, obtained when the coefficients of the squared terms $x^2$ and $y^2$ are positive and different.

Supply Function

An economic function $x = S(p)$ representing the quantity of a commodity $x$ supplied by producers as an increasing function of the unit price $p$.

Demand Function

An economic function $p = D(x)$ (or price function $x = P(p)$) representing the relationship where price $p$ or quantity $x$ demanded decreases as the other variable increases.

Market Equilibrium

The point of intersection between a supply curve and a demand curve, where the quantity supplied equals the quantity demanded at an equilibrium price.

Average Cost Function

The economic function $\bar{C}(x) = \frac{C(x)}{x}$, representing the total production cost $C(x)$ divided by the number of units $x$ produced.

Marginal Function

A function defined as $E(x + 1) - E(x)$, representing the additional change in an economic quantity $E(x)$ resulting from producing or selling one additional unit.

ND (Sign Analysis)

A notation standing for 'not defined', placed on a sign diagram above any $x$-value that makes the denominator of a rational expression equal to zero.

Profit Function

The function $P(x) = R(x) - C(x)$, representing total revenue $R(x)$ minus total cost $C(x)$ for producing and selling $x$ units.