Grade 12 Scientific Mathematics: Calculus, Conics, and Probability Review

Vocabulary flashcards covering fundamental definitions, properties, formulas, and concepts from Grade 12 Advanced High School Mathematics (Calculus, Conic Sections, and Probability).

Antiderivative

A function $F(x)$ defined on an interval $I$ such that $F'(x) = f(x)$ for all $x \text{ in } I$.

Indefinite Integral

The set of all antiderivatives of a function $f(x)$ with respect to $x$, written as $\int f(x)\,dx = F(x) + C$.

Constant of Integration

An arbitrary constant $C$ added to the antiderivative of a function to represent all possible antiderivatives.

Integration by Substitution

A technique for evaluating integrals by defining a new variable $u = g(x)$ with derivative $du = g'(x)\,dx$, rewriting $\int f(g(x))g'(x)\,dx$ as $\int f(u)\,du$.

Integration by Parts

An integration method derived from the product rule of differentiation, expressed by the formula $\int u\,dv = uv - \int v\,du$.

Partial Fractions Decomposition

An algebraic technique used to break down a complex rational function into a sum of simpler fractions to make it easier to integrate.

Definite Integral

The net accumulated value of a continuous function $f(x)$ on an interval $[a, b]$, computed using an antiderivative $F(x)$ as $\int_a^b f(x)\,dx = F(b) - F(a)$.

Volume of Revolution Solid

The volume generated by revolving a region bounded by $y = f(x)$, the x-axis, and lines $x = a$ and $x = b$ completely around the x-axis, given by $V = \pi \int_a^b (f(x))^2\,dx$.

Arc Length

The length of a smooth curve $y = f(x)$ on a closed interval $[a, b]$, calculated as $L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx$.

Differential Equation

An equation that involves an unknown function along with one or more of its derivatives.

Order of a Differential Equation

The order of the highest-order derivative present in the differential equation.

Degree of a Differential Equation

The exponent or power of the highest-order derivative appearing in the differential equation.

Parabola

The set of all points in a plane that are equidistant from a fixed point called the focus $F$ and a fixed line called the directrix.

Ellipse

The set of all points in a plane such that the sum of the distances from two fixed points (foci $F_1$ and $F_2$) is a constant value equal to $2a$.

Hyperbola

The set of all points in a plane such that the absolute difference of the distances from two fixed points (foci $F_1$ and $F_2$) is a constant value equal to $2a$.

Eccentricity

The constant ratio $e = \frac{c}{a}$ of the distance from any point on a conic section to a focus relative to its distance to a directrix.

Random Variable

A function $X: S \rightarrow \mathbb{R}$ that maps outcomes from a sample space $S$ of a random experiment to real numbers.

Discrete Random Variable

A random variable whose range consists of a countable set of distinct real numbers.

Continuous Random Variable

A random variable whose possible values form an uncountable continuous interval of real numbers.

Probability Density Function

A non-negative continuous function $f(x)$ such that the total area under its curve over its domain equals $1$.

Expected Value (Mean) of a Discrete Random Variable

A measure of central tendency for a discrete random variable $X$, defined as $\mu = \sum x_i f(x_i)$.

Cumulative Distribution Function

A function $F(a)$ giving the probability that a random variable $X$ assumes a value less than or equal to $a$, written as $F(a) = P(X \le a)$.

Binomial Distribution

The discrete probability distribution of the number of successes in $n$ independent Bernoulli trials, given by $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$.

Regular (Uniform) Probability Distribution

A continuous probability distribution defined on an interval $[a, b]$ with a constant density function $f(x) = \frac{1}{b-a}$.

Standard Normal Distribution

A continuous, symmetric bell-shaped normal distribution $N(0, 1)$ having a mean $\mu = 0$ and a standard deviation $\sigma = 1$.